Subcomplexes on filtered Riemannian manifolds

Véronique Fischer and Francesca Tripaldi University of Bath, Department of Mathematical Sciences, Bath, BA2 7AY, UK v.c.m.fischer@bath.ac.uk Centro De Giorgi, SNS, Piazza dei Cavalieri 3, 56 126 Pisa, Italy. francesca.tripaldi@sns.it

Abstract.

In this paper, we present a general construction to extract subcomplexes from two distinct complexes on filtered Riemannian manifolds. The first subcomplex computes the de Rham cohomology of the underlying manifold. On regular subRiemannian manifold equipped with a compatible Riemannian metric, it aligns locally with the so-called Rumin complex. The second complex instead generalises the Chevalley-Eilenberg complex computing Lie algebra cohomology of a nilpotent Lie group. Our approach offers key insights on the role of the Riemannian metric when extracting subcomplexes, opening up potential new applications in more general geometric settings, such as singular subRiemannian manifolds.

Key words and phrases:

Differential forms on filtered manifolds, Subcomplexes in sub-Riemannian geometry, Osculating nilpotent Lie groups

2010 Mathematics Subject Classification:

58A10, 58J10, 58H99, 53C17, 43A80, 22E25

1. Introduction

1.1. Motivation

The de Rham complex and its cohomology have inspired many mathematical ideas in the twentieth century, including Hodge theory and the Atiyah-Singer index theorem on Riemannian manifolds. More recently, an increasing number of techniques have been developed in order to extract subcomplexes with the aim of extending such classical results to more general geometric settings [CCY16, Tri54, DH22, Cas11, ČH23, LT23], especially in subRiemannian geometry.

In [FT23], we proposed two constructions on homogeneous nilpotent Lie groups, one by adapting techniques previously developed in the context of parabolic geometry [ČSS01, CD01, DH22] and the other influenced by the ideas in subRiemannian geometry and spectral sequences [Jul95, Rum05]. In the present paper, we push these ideas further to obtain subcomplexes from two different initial complexes and with two equivalent but different constructions. One one hand, we start from the usual de Rham complex $(\Omega^{\bullet}(M),d)$ to extract a subcomplex $(E_{0}^{\bullet},D)$ computing the de Rham cohomology of the underlying manifold $M$ , while on the other, we extract $(\tilde{E}_{0}^{\bullet},\tilde{D})$ from the “algebraic” complex $(\Omega^{\bullet}(M),\tilde{d})$ . Here $\tilde{d}$ denotes the algebraic part of the de Rham differential $d$ , and so the complex $(\Omega^{\bullet}(M),\tilde{d})$ generalises the Chevalley-Eilenberg complex computing the Lie algebra cohomology of a given group to the manifold setting.

1.2. Sketch of the constructions of the complexes

Here, we explain our setting and briefly sketch our two equivalent constructions of each of the two complexes $(E_{0}^{\bullet},D)$ and $(\tilde{E}_{0}^{\bullet},\tilde{D})$ . The various claims will be proved in Section 4.

We consider a filtered manifold $M$ and denote by $\mathfrak{g}_{x}M$ its osculating nilpotent Lie algebra (also called nilpotentisation) above each point $x\in M$ (see Section 3.1). We then define the osculating Chevalley-Eilenberg differential $d_{\mathfrak{g}M}$ on the space $G\Omega^{\bullet}(M)$ of osculating forms over $M$ , defined above each point $x\in M$ as the Chevalley-Eilenberg differential of the osculating group $G_{x}M$ (see Section 3.1.2).

We now further assume that $M$ is also equipped with a metric on $TM$ . This allows us to construct the natural isomorphism $\Phi:\Omega^{\bullet}(M)\to G\Omega^{\bullet}(M)$ , providing an identification between forms and osculating forms (see Section 2.5). We are then able to define the map $d_{0}:=\Phi^{-1}\circ d_{\mathfrak{g}M}\circ\Phi$ acting on $\Omega^{\bullet}(M)$ , which we call the base differential.

1.2.1. Constructing $D$ using $d_{0}^{t}$

The transpose $d_{0}^{t}$ of $d_{0}$ makes sense unambiguously and globally thanks to the metric on $TM$ . We consider the projection $\Pi_{0}$ onto $E_{0}:=\ker d_{0}\cap\ker d_{0}^{t}$ along $F_{0}:={\rm Im}\,d_{0}+{\rm Im}\,d_{0}^{t},$ and the projection $P$ onto $\ker d_{0}^{t}\cap\ker d_{0}^{t}d$ along ${\rm Im}\,d_{0}^{t}+{\rm Im}\,dd_{0}^{t}$ . The proper definitions are in fact given by $\Pi_{0}$ being the orthogonal projection onto the kernel of the algebraic operator $\Box_{0}:=d_{0}d_{0}^{t}+d_{0}^{t}d_{0}$ , and $P$ being the generalised kernel projection of the differential operator $\Box:=dd_{0}^{t}+d_{0}^{t}d$ . Once we define the invertible differential operator $L:=P\Pi_{0}+(\text{\rm I}-P)(\text{\rm I}-\Pi_{0})$ , we have the decomposition

L^{-1}dL=:D+C,

yielding two complexes $(E_{0}^{\bullet},D)$ and $(F_{0}^{\bullet},C)$ whose cohomologies are described as follows:

$\bullet$ $(F_{0}^{\bullet},C)$ has trivial cohomology, since $C$ is conjugated to $d_{0}$ on $F_{0}$ , and $(F_{0}^{\bullet},d_{0})$ is acyclic.

$\bullet$ The cohomology of $(E_{0}^{\bullet},D)$ is linearly isomorphic to the de Rham cohomology of $M$ via the homotopically invertible chain map $\Pi_{0}L^{-1}$ .

1.2.2. Constructing $D$ using $d_{0}^{-1}$

Relying again on the metric on $TM$ , we can define the orthogonal projection ${\rm pr}_{{\rm Im}\,d_{\mathfrak{g}M}}$ onto ${\rm Im}\,d_{\mathfrak{g}M}$ . We then consider the partial inverse $d_{\mathfrak{g}M}^{-1}:=(d_{\mathfrak{g}_{x}M})^{-1}{\rm pr}_{{\rm Im}\,d_{\mathfrak{g}_{x}M}}$ of $d_{\mathfrak{g}_{x}M}$ and thus the corresponding partial inverse $d_{0}^{-1}:=\Phi^{-1}\circ d_{\mathfrak{g}M}^{-1}\circ\Phi$ of $d_{0}$ . Then the projection $P$ above may be constructed explicitly in terms of $d_{0}^{-1}$ as:

P=(\text{\rm I}-b)^{-1}d_{0}^{-1}d+d(\text{\rm I}-b)^{-1}d_{0}^{-1},\quad\mbox{where}\quad b:=d_{0}^{-1}d_{0}-d_{0}^{-1}d.

The complex $D$ is then obtained as $D=\Pi_{0}d(\text{\rm I}-P)\Pi_{0}$ .

1.2.3. Constructions of $\tilde{D}$

We can modify the two constructions above by applying the same steps, but substituting the initial differential operator $d$ with the algebraic part $\tilde{d}$ of the de Rham differential instead (see Section 2.2.1). This yields two complexes $(\tilde{E}_{0}^{\bullet},\tilde{D})$ and $(\tilde{F}_{0}^{\bullet},\tilde{C})$ , with $(\tilde{F}_{0}^{\bullet},\tilde{C})$ acyclic and $(\tilde{E}_{0}^{\bullet},\tilde{D})$ linearly isomorphic to $(\Omega^{\bullet}(M),\tilde{d})$ .

1.3. Novelty and future works

As already mentioned above, constructions related to the ones presented in this paper for the complex $D$ appeared before, in fact more than two decades ago in parabolic geometry [ČSS01, CD01, DH22], and in relation to subRiemannian geometry and spectral sequences [Rum90, Jul95]. Already at that time, it was conjectured that these constructions should align in the resulting subcomplex in some sense (see Section 5.3 in [Rum05], Section 5 in [JK95], Section 2.1 in [Jul19], Section 5.3.3 in [GAJV19], and Section 1.3 in [DH22]), but no proof was offered. An important difficulty incomparing the constructions is that the considerations in the subRiemannian settings were either local or at the level of osculating objects (see Section 5.1 and Remark 5.2 in [Rum05]).

The constructions presented in this paper are set on a filtered Riemannian manifold $M$ ; they yield complexes globally defined and acting on $\Omega^{\bullet}(M)$ - not its osculating counterpart $G\Omega^{\bullet}(M)$ . We obtain two equivalent constructions for the complex $(E_{0}^{\bullet},D)$ , which in the subRiemannian world and together with its osculating couterparts are often referred to as the Rumin complex. We are then able to provide a clearer context and a proof to the conjecture explained above. We also show that these two equivalent constructions may be adapted to yield a different complex, that we have denoted by $(\tilde{E}_{0}^{\bullet},\tilde{D})$ . In Appendix A, we give an explicit construction for the operators $D$ and $\tilde{D}$ on a 3D contact manifold.

We have two main examples in mind for our geometric setting of a filtered Riemannian manifold. We discuss them in turn.

Example 1.1.

A regular subRiemannian manifold equipped with a compatible Riemannian metric (in the sense of Definition 4.13) is naturally a filtered Riemannian manifold.

Within the setting of Example 1.1, we are able to show that the subcomplex $(E_{0}^{\bullet},D)$ coincides with what is now customarily referred to as the Rumin complex on regular $CC$ -structures, first introduced in [Rum90, Rum99, Rum05]. Part of the motivation behind the present work is to shed a new light on Rumin’s construction on regular subRiemannian manifolds and on the nature of constructed objects (e.g. complexes acting on forms or on osculating forms). We also want to address the potential impact that the choice of a Riemannian metric can have on the resulting subcomplex. In particular, in this paper, we emphasise the nontrivial role that the extra hypothesis of compatibility between the Riemannian and subRiemannian structures plays in the construction, see Section 4.6.2. Our hope is to highlight the behaviour of subcomplexes such as the Rumin complex, under changes of variables on regular subRiemannian manifolds [FT12] or pullback by Pansu-differentiable maps on Carnot groups [KMX28].

Example 1.2.

A nilpotent Lie group equipped with a left-invariant filtration is naturally a filtered Riemannian manifold.

Within the setting of Example 1.2, the complex $(\tilde{E}_{0}^{\bullet},\tilde{D})$ computes the Lie algebra cohomology of the given Lie group. A more thorough study of this setting, especially concerning the impact of the particular choice of a filtration on the resulting subcomplex $(E_{0}^{\bullet},D)$ , will be presented in a forthcoming paper [FT24].

We selected the two specific settings in Examples 1.1 and 1.2 due to the exciting recent advances in their applications:

•

large scale geometry (especially the open conjecture classifying nilpotent Lie groups by quasi-isometries [PR18, PT19, BFP22]),
•

analytic torsion [RS12, Kit20, Hal22] and currents [Can21, Vit22, JP12] for subRiemannian geometry, and
•

$K$ and index theories for subelliptic operators and $C^{*}$ -algebras associated with subRiemannian structures [vE10, BvE14].

We believe that the techniques behind the subcomplexes $(E_{0}^{\bullet},D)$ and $(\tilde{E}_{0}^{\bullet},\tilde{D})$ presented in Sections 1.2 and 4 are general enough that their construction will be adaptable to other settings in the future. Interesting generalisations would be to singular subRiemannian manifolds (e.g. Martinet distributions) and to the quasi-subRiemannian setting, for instance the Grushin plane.

1.4. Organisation of the paper

In order to present the hypotheses of our setting and to remove any ambiguity on the objects we consider, the paper starts with some foundational preliminaries on filtrations over vector spaces and on tangent bundles of manifolds (Section 2). When the manifold is filtered (i.e. when the tangent bundle is filtered and the commutator brackets of vector fields respect this filtration), then we can define the osculating Chevalley-Eilenberg differential $d_{\mathfrak{g}M}$ and other osculating objects (see Section 3). In Section 4, we present the general scheme for our construction, discussing the particular case of regular subRiemannian manifold at the end in Section 4.6, and the particular case of 3D contact manifolds in Appendix A.

1.5. Acknowledgements

Both authors acknowledge the support of The Leverhulme Trust through the Research Project Grant 2020-037, Quantum limits for sub-elliptic operators. We are also glad to thank the Centro di Ricerca Matematica Ennio De Giorgi and the Scuola Normale Superiore for the hospitality and support in the early stages of the article, and Professor Giuseppe Tinaglia for hosting us at King’s College London in October 2023.

2. Preliminaries

The main objective of this section is to set some notation for well-known notions regarding manifolds (Section 2.2) and filtrations. For the latter, the filtrations are on vector spaces and vector bundles (Sections 2.1 and 2.4 respectively), which will come in handy when considering a manifold equipped with a metric (Section 2.5).

2.1. Filtrations of vector spaces

Here, we briefly recall the construction of graded objects associated with a filtration of vector spaces. The notions presented below extend naturally to smooth vector bundles and their smooth sections, as well as to modules.

In this paper, all the vector spaces are taken over the field of real numbers.

2.1.1. Filtered vector spaces

A filtered vector space $W$ is a vector space $W$ with a filtration by vector subspaces:

(2.1)

\{0\}=W_{0}\subseteq W_{1}\subseteq\ldots\subseteq W_{s}=W.

In general, a filtration need not be finite; however, in this paper, all the filtrations considered will be finite as above.

A graded vector space $G$ is a vector space $G$ equipped with a gradation by vector subspaces, i.e. a direct sum decomposition by vector subspaces $G_{j}\subset G$ :

(2.2)

\displaystyle G=\oplus_{j=1}^{s}G_{j}\,.

If $G$ is graded, there is a natural filtration associated with its gradation (2.2), given by $\{0\}=W_{0}\subseteq W_{1}\subseteq\cdots\subseteq W_{s}=G$ , where $W_{j}:=\oplus_{k\leq j}G_{k}$ , $j=1,\ldots,s$ .

2.1.2. The vector space ${\rm gr}(W)$

Given a filtered vector space $W$ with the filtration (2.1), its associated graded space is the vector space

(2.3)

{\rm gr}(W):=\oplus_{j=1}^{s}W_{j}/W_{j-1}\,.

Many of the summands above may be trivial. To avoid dealing with zero quotient spaces, it is enough to consider an appropriate choice of indices $w_{0},w_{1},\ldots,w_{s_{0}}\in\mathbb{N}$ for which the summands in (2.3) are not trivial, or equivalently for which the inclusions in (2.1) are strict:

(2.4)

\displaystyle\{0\}=W_{w_{0}}\subsetneq W_{w_{1}}\subsetneq\ldots\subsetneq W_{w_{s_{0}}}=W\ ,\ \mbox{and}\ {\rm gr}\,(W)=\oplus_{j=1}^{s_{0}}W_{w_{j}}/W_{w_{j-1}}.

2.1.3. The linear map ${\rm gr}(T)$

Consider a linear map $T:W\to W^{\prime}$ between two vector spaces, each equipped with a filtration $\{0\}=W_{0}\subseteq W_{1}\subseteq\ldots\subseteq W_{s}=W$ and $\{0\}=W^{\prime}_{0}\subseteq W^{\prime}_{1}\subseteq\ldots\subseteq W^{\prime}_{s^{\prime}}=W^{\prime}$ . We may assume $s=s^{\prime}$ .

The map $T$ is said to respect the filtrations when $T(W_{j})\subseteq W^{\prime}_{j}$ for every $j=0,\ldots,s$ . In this case, one can define the linear map ${\rm gr}(T):{\rm gr}\,(W)\to{\rm gr}\,(W^{\prime})$ as

{\rm gr}(T)\,(v\ {\rm mod}\ W_{j}):=(Tv)\ {\rm mod}\ W^{\prime}_{j}\ \ ,\ v\in W_{j+1}\ ,\ j=0,\ldots,s-1\,.

2.1.4. The basis $\langle e\rangle$

If $W$ is finite dimensional, then ${\rm gr}(W)$ has the same dimension as $W$ . Given $n:=\dim W=\dim{\rm gr}(W)$ and $n_{j}=\dim W_{w_{j}}/W_{w_{j-1}}$ , we say that a basis $e=(e_{1},\ldots,e_{n})$ of $W$ is adapted to the filtration (2.4), when $(e_{1},\ldots,e_{n_{1}})$ is a basis of $W_{w_{1}}$ , $(e_{1},\ldots,e_{n_{2}+n_{1}})$ is a basis of $W_{w_{2}}$ , and so on. For such a basis, we have that, for each $j=0,\ldots,s_{0}-1$ ,

\langle e_{i}\rangle=e_{i}\ {\rm mod}\ W_{w_{j}},\quad i=d_{j}+1,\ldots,d_{j+1}\ ,\text{ with }d_{j}=n_{0}+n_{1}+\cdots+n_{j}\,,

gives a basis $\langle e\rangle:=(\langle e_{1}\rangle,\ldots,\langle e_{n}\rangle)$ of ${\rm gr}(W)$ .

In particular, we get that for each $j=0,\ldots,s_{0}-1$ , the $n_{j}$ -tuple $(\langle e_{d_{j}+1}\rangle,\ldots,\langle e_{d_{j+1}}\rangle)$ is a basis of $W_{w_{j+1}}/W_{w_{j}}$ . In this sense, the basis $\langle e\rangle$ is graded and is said to be the graded basis associated with $e$ .

2.1.5. Matrix representation of linear maps

In this paper, we will often use matrix representations of certain maps. To fix some notation, given a morphism $\varphi:\mathcal{V}_{1}\to\mathcal{V}_{2}$ between two finite dimensional vector spaces, once we fix $\beta_{1}$ and $\beta_{2}$ two bases of $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ respectively, we denote by

{\rm Mat}_{\beta_{1}}^{\beta_{2}}\,\varphi

the matrix representing $\varphi$ in the bases $\beta_{1}$ and $\beta_{2}$ .

Let us take $e$ and $e^{\prime}$ two bases adapted to the filtrations (2.4) of two finite dimensional filtered vector spaces $W$ and $W^{\prime}$ respectively. Assuming $s_{0}=s^{\prime}_{0}$ , $n_{j}=\dim W_{w_{j}}/W_{w_{j-1}}$ and $n^{\prime}_{j}=\dim W^{\prime}_{w_{j}}/W^{\prime}_{w_{j-1}}$ , if a linear map $T\colon W\to W^{\prime}$ respects the filtrations, the matrix representation of $T$ in the bases $e,e^{\prime}$ is block-upper-triangular, i.e.

M:={\rm Mat}_{e}^{e^{\prime}}\,T=\left(\begin{array}[]{ccccc}M_{1}&*&*\\ 0&\ddots&*\\ 0&&M_{s}\\ \end{array}\right),

with $M_{j}\in\mathbb{R}^{n^{\prime}_{j}}\times\mathbb{R}^{n_{j}}$ , with $j=1,\ldots,s_{0}$ . Moreover, the matrix representing ${\rm gr}(T)$ in $\langle e\rangle,\langle e^{\prime}\rangle$ is block-diagonal, i.e.

{\rm gr}(M):={\rm Mat}_{\langle e\rangle}^{\langle e^{\prime}\rangle}\,{\rm gr}(T)=\left(\begin{array}[]{ccccc}M_{1}&0&0\\ 0&\ddots&0\\ 0&&M_{s}\\ \end{array}\right)\,.

2.1.6. Subfiltration

We say that a vector subspace $\tilde{W}\subseteq W$ admits a subfiltration when $\tilde{W}_{w_{j}}:=\tilde{W}\cap W_{w_{j}}$ , $j=1,\ldots,s_{0}$ yields a filtration of $\tilde{W}$ . In this case, the injection map $\tilde{W}\hookrightarrow W$ respects the filtrations and ${\rm gr}\,(\tilde{W})$ is a subspace of ${\rm gr}\,(W)$ .

2.1.7. Decreasing labelling and duality

One can also consider a filtered vector space $W$ with a decreasing filtration, that is a filtration with decreasing labelling:

(2.5)

\displaystyle W=V_{0}\supseteq V_{1}\supseteq\ldots\supseteq V_{t}=\{0\}\,.

It is not difficult to adapt all the constructions considered previously to a decreasing filtration. Indeed, it suffices to change the labels by setting $W_{j}:=V_{t-j}$ , so that, for instance, the associated graded space becomes ${\rm gr}\,(V):=\oplus_{j=0}^{t-1}V_{j}/V_{j+1}$ . However, in this setting, the matrix representations will differ from those in Subsection 2.1.5, as the matrix representing a linear map that respects such decreasing filtrations will be block-lower-triangular.

We observe that if $W$ has a decreasing filtration, then we obtain an increasing filtration of the dual space $W^{\ast}$ of $W$ by considering the filtration by the subspaces $V^{\perp}_{j}\subset W^{\ast}$ , the annihilators of the subspaces $V_{j}\subset W$ , that is

\{0\}=V_{0}^{\perp}\subseteq V_{1}^{\perp}\subseteq\ldots\subseteq V_{t}^{\perp}=W^{\ast}\ ,\ V_{j}^{\perp}=\{\ell\in W^{\ast}\mid\ell|_{V_{j}}\equiv 0\}\,,

also known as the dual filtration of (2.5).

Given two finite dimensional filtered vector spaces $W$ and $W^{\prime}$ , let us take $e$ and $e^{\prime}$ two bases adapted to the decreasing filtrations $W=V_{0}\supseteq V_{1}\supseteq\cdots\supseteq V_{t}=\{0\}$ and $W^{\prime}=V^{\prime}_{0}\supseteq V^{\prime}_{1}\supseteq\cdots\supseteq V^{\prime}_{t}=\{0\}$ . Then for any linear map $T\colon W\to W^{\prime}$ that respects these decreasing filtrations, the dual map

T^{\ast}\colon W^{\prime\ast}\to W^{\ast}\ \text{ where }\ T^{\ast}(\ell)=\ell\circ T\ ,\ \forall\ \ell\in W^{\prime\ast}

respects the dual filtrations, and

\displaystyle{\rm Mat}_{e^{\prime\ast}}^{e^{\ast}}T^{\ast}=\big{(}{\rm Mat}_{e}^{e^{\prime}}T\big{)}^{t}\,,

where $e^{\ast}$ and $e^{\prime\ast}$ denote the dual bases of $e$ and $e^{\prime}$ respectively (one can easily show that if a basis $e$ is adapted to a decreasing filtration (2.5), then $e^{\ast}$ is adapted to its increasing dual filtration). In particular, this readily implies that the matrix representation of $T^{\ast}\colon W^{\prime\ast}\to W^{\ast}$ is block-upper-triangular, as pointed out in Subection 2.1.5.

Moreover, the vector spaces ${\rm gr}(V^{*})$ and $({\rm gr}\,(V))^{*}$ are canonically isomorphic.

2.1.8. Graded vector spaces

As explained earlier, given a graded vector space $G$ , its gradation (2.2) naturally produces a filtration through $\oplus_{k\leq j}G_{k}$ , while a filtration (2.1) of a vector space $W$ yields a graded space ${\rm gr}\,(W)$ (2.3).

Definition 2.1.

Let $G=\oplus_{j=1}^{s}G_{j}$ and $G^{\prime}=\oplus_{j^{\prime}=1}^{s^{\prime}}G^{\prime}_{j}$ be two graded vector spaces. We may assume $s=s^{\prime}$ . A linear map $T:G\to G^{\prime}$ respects the gradations when $T(G_{j})\subset G^{\prime}_{j}$ for any $j=1,\ldots,s$ .

For instance, if $T$ is a linear map between two filtered vector spaces that respects the filtrations, then ${\rm gr}\,(T)$ respects the gradations.

Consider a graded vector space $G$ as above. Then its dual $G^{*}$ is also graded with the dual gradation $G^{*}=\oplus_{j=1}^{s}G_{j}^{\perp}$ . The $k^{th}$ exterior algebra is also graded via the (possibly infinite) gradation

\wedge^{k}G=\oplus_{w=1}^{\infty}G^{k,w}\,\text{, where }\ G^{k,w}:=\sum_{j_{1}+\cdots+j_{k}=w}G_{j_{1}}\wedge\cdots\wedge G_{j_{k}}\,.

This leads to a (possibly infinite) gradation of the exterior algebra $\wedge^{\bullet}G$ which respects the wedge product:

\wedge^{\bullet}G=\oplus_{w=0}^{\infty}G^{\bullet,w},\qquad G^{\bullet,w}=\oplus_{k=0}^{\infty}G^{k,w},

with $\wedge^{0}G=G^{0,0}=\mathbb{R}$ , and $G^{0,w}=\{0\}$ for $w>0$ . If $G$ is finite dimensional, the gradations of $\wedge^{k}G$ and $\wedge^{\bullet}G$ are finite.

In the graded setting, there is a natural notion of weight: an element in $G_{j}$ has weight $j$ , and more generally, an element in $G^{\bullet,w}$ has weight $w$ .

2.1.9. Filtration on the exterior algebra.

A filtration (2.1) on a vector space $W$ also induces a decreasing filtration on its $k^{th}$ exterior algebra:

(2.6)

\wedge^{k}W=W^{k,\geq 0}\supseteq W^{k,\geq 1}\supseteq\ldots\supseteq W^{k,\geq k\cdot s+1}=\{0\},

where

W^{k,\geq w}:=\sum_{j_{1}+\cdots+j_{k}\geq w}W_{j_{1}}\wedge\cdots\wedge W_{j_{k}}\,.

The following lemma is easily checked.

Lemma 2.2.

Let $W$ be a finite dimensional vector space equipped with a filtration (2.1), and let $e=(e_{1},\ldots,e_{n})$ be a basis of $W$ adapted to the filtration. Define the linear map

\Psi:\wedge^{k}W\to\wedge^{k}{\rm gr}\,(W),

via

\Psi(e_{i_{1}}\wedge\cdots\wedge e_{i_{k}})=\langle e_{i_{1}}\rangle\wedge\cdots\wedge\langle e_{i_{k}}\rangle,\quad i_{1}<\ldots<i_{n}.

Then $\Psi$ is a linear isomorphism that respects the filtration (2.6) of $\wedge^{k}W$ and the one induced by the gradation of $\wedge^{k}{\rm gr}\,(W)$ . Moreover, ${\rm gr}\,(\Psi)$ provides an isomorphism between

{\rm gr}(\wedge^{k}W)=\oplus_{w=0}^{k\cdot s}W^{k,\geq w}/W^{k,\geq w+1}

and $\wedge^{k}{\rm gr}\,(W)$ respecting their gradations.

Note that the isomorphism $\Psi$ depends on the choice of the basis $e$ .

2.1.10. Euclidean filtered vector spaces

Here, we consider a Euclidean space, that is, a finite dimensional vector space $W$ equipped with a scalar product $(\cdot,\cdot)_{W}$ .

Any subspace $W^{\prime}\subset W$ is naturally Euclidean, its scalar product $(\cdot,\cdot)_{W^{\prime}}$ being obtained by restricting $(\cdot,\cdot)_{W}$ . Moreover, its orthogonal complement $W^{\prime(\perp,W)}$ in $W$ is naturally identified with the quotient

W/W^{\prime}\cong W^{\prime(\perp,W)}.

This quotient $W/W^{\prime}$ is also Euclidean if we equip it with the scalar product $(\cdot,\cdot)_{W/W^{\prime}}$ corresponding to $(\cdot,\cdot)_{W^{\prime(\perp,W)}}$ .

Beside being Euclidean, we further assume the vector space $W$ also be filtered with filtration (2.4). The resulting graded space ${\rm gr}\,W$ inherits a natural scalar product $(\cdot,\cdot)_{{\rm gr}W}$ by imposing that the decomposition (2.3) is orthogonal, and that $(\cdot,\cdot)_{{\rm gr}W}$ restricted to each $W_{w_{j}}/W_{w_{j-1}}$ is given by $(\cdot,\cdot)_{W_{w_{j}}/W_{w_{j-1}}}$ . Indeed, each $W_{w_{j}}/W_{w_{j-1}}$ is naturally isomorphic to the orthogonal complement $V_{j}:=W_{n_{j-1}}^{(\perp,W_{n_{j}})}$ of $W_{w_{j-1}}$ in $W_{w_{j}}$ . Therefore, by construction, the gradation by quotients on ${\rm gr}\,(W)$ is isomorphic to the following gradation of $W$ :

W=V_{1}\oplus^{\perp}V_{2}\oplus^{\perp}\ldots\oplus^{\perp}V_{s_{0}},

which we will refer to as the orthogonal gradation or orthogonal graded decomposition of $W$ (associated with the given scalar product and filtration).

We observe that when considering a basis $e=(e_{1},\ldots,e_{n})$ of $W$ adapted to its filtration, after a Graham-Schmidt process, we may assume that $(e_{1},\ldots,e_{n_{1}})$ is an orthonormal basis of $V_{1}=W_{1}$ , $(e_{n_{1}+1},\ldots,e_{n_{2}+n_{1}})$ is an orthonormal basis of $V_{2}$ , etc. In other words, after applying a Graham-Schmidt process, we can always obtain a basis $e$ adapted to the given orthogonal gradation of $W$ .

2.2. Preliminaries on manifolds

2.2.1. The maps $d$ and $\tilde{d}$ on a general manifold

In this section, we recall briefly some well-known facts about the de Rham differential which are valid on any smooth manifold $M$ (without any further assumptions). As is customary, $d$ denotes the exterior differential on the space of smooth forms $\Omega^{\bullet}(M)$ on $M$ . It is well known that the map $d\colon\Omega^{k}(M)\to\Omega^{k+1}(M)$ is a smooth differential map that satisfies the Leibniz property and $d^{2}=0$ . The associated cochain complex $(\Omega^{\bullet}(M),d)$ is traditionally referred to as the de Rham complex.

The explicit formula for $d$ is given for any $\omega\in\Omega^{k}(M)$ and $V_{0},\ldots,V_{k}\in\Gamma(TM)$ by

	$\displaystyle d\omega(V_{0},\ldots,V_{k})$	$\displaystyle=\sum_{i=0}^{k}(-1)^{i}V_{i}\left(\omega(V_{0},\ldots,\hat{V}_{i},\ldots,V_{k})\right)$
		$\displaystyle\qquad+\sum_{0\leq i<j\leq k}(-1)^{i+j}\omega([V_{i},V_{j}],V_{0},\ldots,\hat{V}_{i},\ldots,\hat{V}_{j},\ldots,V_{k}),$

the hats denoting omissions.

In this paper, the algebraic part of $d$ is denoted by $\tilde{d}$ . This is the map given by $\tilde{d}=0$ on $\Omega^{0}(M)$ , and for $k>0$ and any $\omega\in\Omega^{k}(M)$ , $V_{0},\ldots,V_{k}\in\Gamma(TM)$ , by

(2.7)

\tilde{d}\omega(V_{0},\ldots,V_{k})=\sum_{0\leq i<j\leq k}(-1)^{i+j}\omega([V_{i},V_{j}],V_{0},\ldots,\hat{V}_{i},\ldots,\hat{V}_{j},\ldots,V_{k}).

Lemma 2.3.

The map $\tilde{d}:\Omega^{k}(M)\to\Omega^{k+1}(M)$ is smooth and algebraic. It satisfies the Leibniz property and $(\Omega^{\bullet}(M),\tilde{d})$ is a complex, i.e. $\tilde{d}^{2}=0$ . The action of $\tilde{d}$ over $\Omega^{1}(M)$ is given by the explicit formula

\tilde{d}\omega(V_{0},V_{1})=-\omega([V_{0},V_{1}])\ ,\ \forall\,\omega\in\Omega^{1}(M),\ V_{0},V_{1}\in\Gamma(TM)\,.

Sketch of the proof.

Since $\tilde{d}=d-(d-\tilde{d})$ and for any $\omega\in\Omega^{k}(M),\,V_{0},\ldots,V_{k}\in\Gamma(TM)$

(2.8)

(d-\tilde{d})\,\omega(V_{0},\ldots,V_{k})=\sum_{i=0}^{k}(-1)^{i}V_{i}\left(\omega(V_{0},\ldots,\hat{V}_{i},\ldots,V_{k})\right)\ ,

the Leibniz property of $\tilde{d}$ follows from the fact that both vector fields and $d$ satisfy the Leibniz property.

The formula on 1-forms is easily computed from (2.7) by taking $k=1$ . This formula, together with the Jacobi identity for the commutator bracket, implies $\tilde{d}^{2}=0$ on $\Omega^{1}(M)$ . Recursively and by the Leibniz rule, we get that $\tilde{d}^{2}=0$ on forms of any degree, since for any $\alpha\in\Omega^{k}(M)$ and any $\beta\in\Omega^{\bullet}(M)$

	$\displaystyle\tilde{d}^{2}(\alpha\wedge\beta)=$	$\displaystyle\tilde{d}\big{(}\tilde{d}\alpha\wedge\beta+(-1)^{k}\alpha\wedge\tilde{d}\beta\big{)}$
	$\displaystyle=$	$\displaystyle(\tilde{d}^{2}\alpha)\wedge\beta+(-1)^{k+1}\tilde{d}\alpha\wedge\tilde{d}\beta+(-1)^{k}\tilde{d}\alpha\wedge\tilde{d}\beta+\alpha\wedge\tilde{d}^{2}\beta=0\,.$

∎

2.3. Some natural concepts

In this subsection, we recall some basic concepts in differential geometry.

2.3.1. Frame and coframe

Let $E$ be a (real, finite dimensional, and smooth) vector bundle over a (smooth) manifold $M$ with $\dim M=n$ . A frame for $E$ is a smooth section $(S_{1},\ldots,S_{n})$ of $E\times\ldots\times E=E^{n}$ such that $(S_{1}(x),\ldots,S_{n}(x))$ is a basis of $E_{x}$ for every $x\in M$ . A coframe for $E$ is frame for the dual vector bundle $E^{*}$ .

Throughout this paper, if the bundle $E$ is not specified, we will take $E$ to be the tangent bundle $TM$ over the smooth manifold $M$ and $E^{*}$ to be the cotangent bundle. Frames for $TM$ may not exist globally on the whole manifold $M$ , but they can always be constructed locally, i.e. over an open neighbourhood of every point in $M$ .

2.3.2. Algebraic maps on bundles

Let $E$ and $F$ be two smooth vector bundles over a manifold $M$ . We say that a smooth map $T:\Gamma(E)\to\Gamma(F)$ is linear and algebraic when, for each $x\in M$ , there exists a linear map $(T)_{x}:E_{x}\to F_{x}$ satisfying $T(\alpha)(x)=(T)_{x}(\alpha(x))$ for any $\alpha\in E$ , $x\in M$ . We will call $(T)_{x}$ the pointwise restriction of $T$ above $x\in M$ .

Remark 2.4.

A (smooth) differential operator of order 0 is an algebraic linear map.

2.3.3. Metric on bundles

Let $E$ be a vector bundle over a manifold $M$ . By definition, a metric $g$ on $E$ is a family of scalar products $g_{x}:=(\cdot,\cdot)_{E_{x}}$ on $E_{x}$ depending smoothly on $x\in M$ . The smooth dependence means that

g\in\Gamma({\rm Sym}(E\otimes E))\,.

When $E$ is the tangent bundle $TM$ of a smooth manifold $M$ , the metric is said to be Riemannian and the couple $(M,g)$ is referred to as a Riemannian manifold.

2.4. Filtered tangent bundles

In this section, we consider an $n$ -dimensional smooth manifold $M$ whose tangent bundle $TM$ is filtered by vector subbundles

(2.9)

M\times\{0\}=H^{0}\subseteq H^{1}\subseteq\ldots\subseteq H^{s}=TM\,.

Our aim is to extend the concepts of Section 2.1 to this setting. After an appropriate choice of indices like in (2.4), one can define the smooth vector bundle

{\rm gr}(TM)=\oplus_{i=1}^{s}H^{i}/H^{i-1}=\oplus_{j=1}^{s_{0}}H^{w_{j}}/H^{w_{j-1}}\ ,

with $n_{j}:=\dim H^{w_{j}}/H^{w_{j-1}}>0,\ j=1,\ldots,s_{0}$ , and $dim(M)=n=n_{1}+n_{2}+\cdots+n_{s_{0}}$ .

2.4.1. Adapted frames

The notion of adapted bases in the setting of filtered vector spaces naturally leads to the notion of adapted frames and coframes.

Definition 2.5.

A frame $\mathbb{X}=(X_{1},\ldots,X_{n})$ for $TM$ is said to be adapted to the filtration (2.9) when, for every $x\in M$ , the basis $(X_{1}(x),\ldots,X_{n}(x))$ is adapted to the filtration $T_{x}M=H^{s_{0}}_{x}\supseteq\ldots\supseteq H^{w_{1}}_{x}\supseteq H^{w_{0}}_{x}=\{0\}$ .

If $\mathbb{X}=(X_{1},\ldots,X_{n})$ is a frame for $TM$ adapted to the filtration (2.9), then the associated graded basis on ${\rm gr}\,(T_{x}M)$

\langle\mathbb{X}\rangle_{x}=(\langle X_{1}\rangle_{x},\ldots,\langle X_{n}\rangle_{x})\,,\ x\in M\,,

yields a frame $\langle\mathbb{X}\rangle$ for ${\rm gr}(TM)$ adapted to the gradation in the following sense.

Definition 2.6.

A frame $\mathbb{Y}=(Y_{1},\ldots,Y_{n})$ for ${\rm gr}(TM)$ is adapted to the gradation, or graded, when $Y_{1},\ldots,Y_{n_{1}}$ is a frame for $H^{w_{1}}/H^{w_{0}}$ , $Y_{n_{1}+1},\ldots,Y_{n_{2}+n_{1}}$ is a frame for $H^{w_{2}}/H^{w_{1}}$ , and so on.

By duality, we obtain the following decreasing filtration of the cotangent bundle $T^{*}M$ by vector bundles

(2.10)

T^{*}M=(H^{w_{0}})^{\perp}\supseteq(H^{w_{1}})^{\perp}\supseteq\ldots\supseteq(H^{w_{s_{0}}})^{\perp}=M\times\{0\}\,,

by considering the annihilators $(H^{w_{j}})^{\perp}$ of $H^{w_{j}}$ for $j=1,\ldots,s_{0}$ , as follows.

Definition 2.7.

A coframe $\Theta$ for $TM$ is adapted to the filtration when, for every $x\in M$ , the basis $\Theta(x):=(\theta^{1}(x),\ldots,\theta^{n}(x))$ is adapted to the filtration of $T_{x}^{*}M=(H^{w_{0}}_{x})^{\perp}\supseteq(H^{w_{1}}_{x})^{\perp}\supseteq\ldots\supseteq(H^{w_{s_{0}}}_{x})^{\perp}=\{0\}$ .

This definition is equivalent to saying that, at each point $x\in M$ , the final $n_{s_{0}}$ covectors annihilate $H^{w_{s_{0}-1}}_{x}$ . Furthermore, these final $n_{s_{0}}$ covectors together with the preceding $n_{s_{0}-1}$ covectors annihilate $H^{s_{0}-2}_{x}$ , and so on. The associated frame $\langle\Theta\rangle=(\langle\theta^{1}\rangle,\ldots,\langle\theta^{n}\rangle)$ is a frame for ${\rm gr}(T^{*}M)$ that is adapted to the gradation in the following sense.

Definition 2.8.

A frame $\Xi=(\xi^{1},\ldots,\xi^{n})$ for ${\rm gr}(T^{*}M)$ is said to be adapted to the gradation (or simply graded) when $\xi^{n},\ldots,\xi^{n-n_{s_{0}}+1}$ is a frame for $(H^{w_{s_{0}-1}})^{\perp}/(H^{w_{s_{0}}})^{\perp}$ , $\xi^{n-n_{s_{0}}},\ldots,$ $\xi^{n-n_{s_{0}-1}+1}$ is a frame for $(H^{w_{s_{0}-2}})^{\perp}/(H^{w_{s_{0}-1}})^{\perp}$ , and so on.

One can readily check that if $\mathbb{X}$ is a frame adapted to a gradation for $TM$ , then its dual $\Theta$ is an adapted coframe for $TM$ . Conversely, if $\Theta$ is a coframe for $T^{\ast}M$ adapted to a gradation of $TM$ , then its dual $\Theta^{\ast}=\mathbb{X}$ is a graded frame for $TM$ .

As mentioned in Subsection 2.3.1, frames for $TM$ can always be constructed locally. Furthermore, through a pivoting process, one may easily construct a local frame for $TM$ adapted to a given gradation. The inverse function theorem implies that, given a local frame $\mathbb{Y}$ of ${\rm gr}(TM)$ adapted to the gradation, we can then construct a local frame $\mathbb{X}$ of $TM$ adapted to the filtration such that $\langle\mathbb{X}\rangle=\mathbb{Y}$ .

2.4.2. Weights of forms

Building on top of Section 2.4.1, we define below the natural notion of forms of weights at least $w$ in this context.

Given an increasing filtration (2.9) of $TM$ , we denote by $H^{j}$ the set of (based) vectors of weight at most $j$ (shortened with $\leq j$ ). By duality, we say that the (based) covectors in $(H^{j-1})^{\perp}$ have weight at least $j$ (shortened as $\geq j$ ). We extend this vocabulary to the space of smooth sections $\Gamma(TM)$ and $\Gamma(T^{*}M)=\Omega^{1}(M)$ , that is vector fields and 1-forms respectively.

With our definition, a 1-form $\alpha\in\Omega^{1}(M)$ is of weight $\geq w$ when $\alpha(V)=0$ for any $V\in\Gamma(H^{w-1})$ . We denote the space of 1-forms of weight at least $j$ by

\Omega^{1,\geq j}(M):=\Gamma((H^{j-1})^{\perp}),\qquad j=1,2,\ldots,s_{0}+1\,.

The filtration in (2.10) of the cotangent bundle then determines the following strict filtration

\Omega^{1}(M)=\Omega^{1,\geq w_{1}}(M)\supsetneq\Omega^{1,\geq w_{2}}(M)\supsetneq\ldots\supsetneq\Omega^{1,\geq w_{s_{0}+1}}(M)=\{0\}.

Example 2.9.

If $\Theta:=(\theta^{1},\ldots,\theta^{n})$ is a frame for $T^{\ast}M$ adapted to the filtration (2.10), then all the 1-forms $\theta^{1},\ldots,\theta^{n}$ are of weight $\geq w_{1}$ . The 1-forms $\theta^{n_{1}+1},\ldots,\theta^{n}$ are of weight $\geq w_{2}$ , and more generally $\theta^{n_{1}+\cdots+n_{i-1}+1},\ldots,\theta^{n}$ are of weight $\geq w_{i}$ . By taking $n_{0}=\dim H^{w_{0}}=0$ , we can state this as

\theta^{j}\in\Omega^{1,\geq w_{i}}(M),\quad j=n_{0}+\cdots+n_{i-1}+1,\ldots,n\ ,\quad i=1,\ldots,s_{0}+1.

In general, we will not have a global coframe as in Example 2.9, but just local ones.

For $k=0$ , we set

C^{\infty}(M)=\Omega^{0}(M):=\Omega^{0,\geq 0}(M),\qquad\mbox{and}\qquad\Omega^{0,\geq 1}(M):=\{0\}.

For $k=1,2,\ldots$ , we define the space of $k$ -forms of weight at least $w$ (shortened as $\geq w$ ) by

\Omega^{k,\geq w}(M):=\oplus_{i_{1}+\ldots+i_{k}\geq w}\Omega^{1,\geq i_{1}}(M)\wedge\ldots\wedge\Omega^{1,\geq i_{k}}(M).

In other words, a $k$ -form is of weight at least $w$ when it can be written locally as a linear combination of $k$ -wedges of 1-form of weights at least $i_{1},\ldots,i_{k}$ with $w\leq i_{1}+\ldots+i_{k}$ . From the definition, it follows that a $k$ -form $\alpha\in\Omega^{k}(M)$ is of weight $\geq w$ when

\forall\ V_{1}\in\Gamma(H^{i_{1}}),\ldots,V_{k}\in\Gamma(H^{i_{k}})\qquad i_{1}+\ldots+i_{k}<w\Longrightarrow\alpha(V_{1},\ldots,V_{k})=0.

Example 2.10.

Consider an adapted coframe $\Theta$ for $TM$ . We will often use the shorthand

\Theta^{\wedge I}=\theta^{i_{1}}\wedge\ldots\wedge\theta^{i_{k}}

for the multi-index $I=(i_{1},\ldots,i_{k})$ , always assuming $i_{1}<\ldots<i_{k}$ . Then $\Theta^{\wedge I}$ is of weight $\geq\upsilon_{1}+\ldots+\upsilon_{k}$ , where each $\theta^{i_{j}}$ is of weight $\geq\upsilon_{j}$ .

Definition 2.11.

The family $(\Theta^{I})_{I}$ where $I$ runs above all multi-indices of length $k$ (taken with strictly increasing indices as in example 2.10 above) is a free basis of the $C^{\infty}(M)$ -module $\Omega^{\bullet}(M)$ called the basis associated with the coframe $\Theta$ .

By construction, we have the inclusion $\Omega^{k,\geq w}(M)\wedge\Omega^{j,\geq w^{\prime}}(M)\subseteq\Omega^{k+j,\geq w+w^{\prime}}(M)$ for any $k,j\geq 0$ . Moreover, given the increasing filtration (2.4) of $TM$ , for any degree $k\geq 0$

\Omega^{k,\geq k\cdot w_{s_{0}}+1}(M)=\{0\}\ \text{ and }\ w\leq w^{\prime}\ \Longrightarrow\ \Omega^{k,\geq w}(M)\supseteq\Omega^{k,\geq w^{\prime}}(M)\,.

In particular, the module $\Omega^{k}(M)$ over the ring $C^{\infty}(M)$ admits the filtration

(2.11)

\displaystyle\Omega^{k}(M)=\Omega^{k,\geq 0}(M)\supset\Omega^{k,\geq 1}(M)\supset\ldots\supset\Omega^{k,\geq w}(M)\supset\ldots\supset\Omega^{k,\geq k\cdot w_{s_{0}}+1}(M)=\{0\}.

The basis $(\Theta^{I})_{I}$ introduced in Definition 2.11 is adapted to this filtration.

2.4.3. The bundle $\wedge^{k}{\rm gr}(T^{*}M)$ and its weights

Given the filtration (2.9), by duality the smooth vector bundle

{\rm gr}(T^{*}M)=\oplus_{i=1}^{s}((H^{i-1})^{\perp}/(H^{i})^{\perp})=\oplus_{j=1}^{s_{0}}((H^{w_{j}-1})^{\perp}/(H^{w_{j}})^{\perp})

is graded. We say that an element of ${\rm gr}(T^{*}M)$ is of weight $j$ when it is in $(H^{j-1})^{\perp}/(H^{j})^{\perp}$ . We extend the same notion of weight to smooth sections of ${\rm gr}(T^{*}M)$ .

Example 2.12.

Let us consider the same adapted coframe $\Theta$ for $TM$ as in Example 2.9. Then $\langle\theta^{1}\rangle,\ldots,\langle\theta^{n_{1}}\rangle$ have weight $w_{1}$ , and for $i=2,\ldots,s_{0}$ , the sections $\langle\theta^{n_{1}+\cdots+n_{i-1}+1}\rangle,\ldots,$ $\langle\theta^{n_{1}+\cdots+n_{i}}\rangle$ of ${\rm gr}(T^{*}M)$ have weight $w_{i}$ . If we denote by $\upsilon_{j}$ the weight of $\langle\theta^{j}\rangle$ , we have:

(2.12)

\upsilon_{n_{1}+\cdots+n_{i-1}+1}=\ldots=\upsilon_{n_{1}+\cdots+n_{i}}=w_{i},\qquad i=1,\ldots,s_{0}.

There is a natural identification between the following smooth vector bundles

\wedge^{k}{\rm gr}(T^{*}M):=\cup_{x\in M}\wedge^{k}{\rm gr}(T_{x}^{*}M)\cong\cup_{x\in M}(\wedge^{k}{\rm gr}(T_{x}M))^{*}=:\wedge^{k}{\rm gr}(TM)^{*},

allowing us to consider directly $\wedge^{k}{\rm gr}(T^{*}M)$ in our discussion.

It is straightforward to check that $\wedge^{k}{\rm gr}(T^{*}M)$ is a module over $C^{\infty}(M)$ which, by Section 2.1.8, is naturally equipped with a gradation and a notion of weight. In other words,

\wedge^{k}{\rm gr}(T^{*}M)=\oplus_{w\in\mathbb{N}}\wedge^{k,w}{\rm gr}(T^{*}M),

where the elements of weight $w$ are in the linear subbundle

\wedge^{k,w}{\rm gr}(T^{*}M):=\sum_{i_{1}+\ldots+i_{k}=w}((H^{i_{1}-1})^{\perp}/(H^{i_{1}})^{\perp})\wedge\ldots\wedge((H^{i_{k}-1})^{\perp}/(H^{i_{k}})^{\perp}).

Once we extend this construction to the space of smooth sections of $\wedge^{\bullet}{\rm gr}\,(T^{\ast}M)$ , we get the gradation

(2.13)

\displaystyle\Gamma(\wedge^{\bullet}{\rm gr}\,(T^{\ast}M))=\oplus_{k,w\in\mathbb{N}_{0}}\ \Gamma(\wedge^{k,w}{\rm gr}\,(T^{\ast}M))\,,

where the $k$ -forms of weight $w$ are given by

\displaystyle\Gamma(\wedge^{k,w}{\rm gr}\,(T^{\ast}M))=

\displaystyle\sum_{i_{1}+\cdots+i_{k}=w}\Gamma((H^{i_{1}-1})^{\perp})/\Gamma((H^{i_{1}})^{\perp})\wedge\cdots\wedge\Gamma((H^{i_{k}-1})^{\perp})/\Gamma((H^{i_{k}})^{\perp})\,.

Example 2.13.

We continue with the setting of Example 2.12 and examine the case of $k$ -forms, $k>1$ . The sections $\langle\theta^{i_{1}}\rangle\wedge\ldots\wedge\langle\theta^{i_{k}}\rangle$ of $\wedge^{k}{\rm gr}(T^{*}M)$ are of weight

|(\upsilon_{1},\ldots,\upsilon_{k})|:=\upsilon_{i_{1}}+\ldots+\upsilon_{i_{k}}.

We may use the shorthand notation $\langle\Theta\rangle^{\wedge I}=\langle\theta^{i_{1}}\rangle\wedge\ldots\wedge\langle\theta^{i_{k}}\rangle$ where $I$ is the multi-index $I=(i_{1},\ldots,i_{k})$ with $i_{1}<\ldots<i_{k}$ .

We readily check the following properties:

Lemma 2.14.

The family $(\langle\Theta\rangle^{I})_{I}$ , where $I$ runs above all multi-indices of length $k$ and strictly increasing indices, is a basis of $\Gamma(\wedge^{k}{\rm gr}(T^{*}M))$ adapted to the gradation. Moreover, it is the basis associated to the basis $(\Theta^{I})_{I}$ adapted to the filtration (2.11) of $\Omega^{\bullet}(M)$ in the sense described in Section 2.1.4.

2.4.4. The ${\rm gr}$ operation

Definition 2.15.

A morphism $D:\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ of $C^{\infty}(M)$ -modules respects the filtration (2.11), when

\forall w\in\mathbb{N}_{0}\qquad D\,\Omega^{\bullet,\geq w}(M)\subseteq\Omega^{\bullet,\geq w}(M).

The following property follows readily from Section 2.1:

Lemma 2.16.

Let $D:\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ be a morphism of $C^{\infty}(M)$ -modules that respects the filtration (2.11). Then the map

{\rm gr}(D):\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))\to\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M)),

is a morphism of $C^{\infty}(M)$ -modules that respects the gradation (2.13). If in addition, $D$ is a differential operator, then ${\rm gr}(D)$ is also a differential operator of the same order or lower.

As an operation, ${\rm gr}$ is a module morphism from $\{D:\Omega^{\bullet}(M)\to\Omega^{\bullet}(M):$ module morphism $\}$ to $\{T:\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))\to\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M)):$ module morphism $\}$ .

By construction, ${\rm gr}(D)=0$ if and only if $D$ increases the weight in the following sense.

Definition 2.17.

We say that a linear map $D:\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ increases the weights of the filtration when

\forall w\in\mathbb{N}_{0}\qquad D\,\Omega^{\bullet,\geq w}(M)\subseteq\Omega^{\bullet,\geq w+1}(M).

As there are only a finite number of weights, a linear map that increases the weights is necessarily nilpotent, that is

(2.14)

{\rm gr}\,D=0\Longrightarrow\quad\exists\,N_{0}\in\mathbb{N}\ \text{ such that }D^{N_{0}}=0.

This $N_{0}$ depends only on $M$ and its filtration (2.9) (via the filtrations (2.11)), not on $D$ . Throughout this paper, we will assume it to be a fixed integer.

2.5. Tangent bundles equipped with a filtration and a metric

Let $M$ be a manifold whose tangent bundle admits a filtration (2.9) by subbundles and is equipped with a metric $g_{TM}$ .

2.5.1. Identification of $\Omega^{\bullet}(M)$ and $\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$

The considerations in Section 2.1.10 show that ${\rm gr}(TM)$ inherits a metric $g_{{\rm gr}(TM)}$ . The same section also implies that, at each point $x\in M$ , there exists an open neighbourhood $U$ of $x$ and a frame $\mathbb{X}$ of $TM$ on $U$ that is $g_{TM}$ -orthonormal and adapted to the natural orthogonal gradation

TM=\oplus_{j}^{\perp}G^{j},

where $G^{j}$ is the orthogonal complement of $H^{w_{j-1}}$ in $H^{w_{j}}$ for each $j=1,\ldots,s_{0}$ .

Let $x\in M$ . For any $j=1,\ldots,s_{0}$ , if $\xi$ is a covector in $G^{j}_{x}$ (i.e. a linear form on $T_{x}M$ vanishing on every $G^{k}_{x}$ , $k\neq j$ ), we denote by $\Phi_{x}(\xi)$ the corresponding covector in $H^{w_{j}}_{x}/H^{w_{j-1}}_{x}$ (i.e. the corresponding linear form on ${\rm gr}(T^{*}_{x}M)$ vanishing on $H^{w_{k}}_{x}/H^{w_{k-1}}_{x}$ , $k\neq j$ ). This extends uniquely to a linear bijective map $\Phi_{x}:T_{x}^{*}M\to{\rm gr}(T_{x}^{*}M)$ , and then to a $C^{\infty}(M)$ -module isomorphism

\Phi:\Omega^{\bullet}(M)\to\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))

respecting the Leibniz property.

Lemma 2.18.

Let $M$ be a manifold whose tangent bundle admits a filtration by subbundles (2.9) and is equipped with a metric $g_{TM}$ . The map $\Phi$ constructed above is an isomorphism of $C^{\infty}(M)$ -modules that respects the filtrations, even when restricted to $\Omega^{k}(M)\to\Gamma(\wedge^{k}{\rm gr}(T^{*}M))$ for each $k=0,1,\ldots,n$ . Moreover, ${\rm gr}(\Phi)$ is an isomorphism between the $C^{\infty}(M)$ -modules ${\rm gr}(\Omega^{k}(M))$ and $\Gamma(\wedge^{k}{\rm gr}(T^{*}M))$ that respects the gradation.

If $\mathbb{X}$ is a $g_{TM}$ -orthonormal local frame of $TM$ on an open subset $U\subset M$ that is adapted to the natural orthogonal gradation $TM=\oplus^{\perp}G^{j}$ , then we have

\Phi(\Theta^{I})=\langle\Theta\rangle^{I}

for any increasing multi-index $I$ of length $k$ , where $\Theta$ is the coframe associated to $\mathbb{X}$ over $U$ .

Note that the map $\Phi$ is in fact algebraic, with

\Phi_{x}:\Gamma(\wedge^{\bullet}T_{x}^{*}M)\to\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))\quad\mbox{defined via}\quad\Phi_{x}(\Theta^{I}(x))=\langle\Theta\rangle^{I}(x),

using the notation of Lemma 2.18. When restricted to $\Omega^{1}(M)$ , the matrix representing the map $\Phi$ is just the identity matrix of $\mathbb{R}^{n}$ :

{\rm Mat}_{\Theta}^{\langle\Theta\rangle}(\Phi:\Omega^{1}(M)\to\Gamma({\rm gr}(T^{\ast}M)))=\text{\rm I}.

Moreover, for each $k=2,\ldots,n$ , using the notation $\Theta^{k}=(\Theta^{I})_{I}$ and $\langle\Theta\rangle^{k}=(\langle\Theta\rangle^{I})_{I}$ for the corresponding bases of $\Omega^{k}(M)$ and of $\Gamma(\wedge^{k}{\rm gr}(T^{*}M))$ , the matrix representing $\Phi\colon\Omega^{k}(M)\to\Gamma(\wedge^{k}{\rm gr}(T^{\ast}M))$ is the identity matrix of $\mathbb{R}^{\dim\Omega^{k}(M)}$ :

{\rm Mat}_{\Theta^{k}}^{\langle\Theta\rangle^{k}}(\Phi)=\text{\rm I}\,.

Intertwining with $\Phi$ provides forms with a reverse operation to the operation ${\rm gr}$ described in Lemma 2.16:

Lemma 2.19.

We continue with the setting of Lemma 2.18.

(1)

If $T:\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))\to\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$ is a morphism of $C^{\infty}(M)$ -modules, then

$T^{\Phi}:=\Phi^{-1}\circ T\circ\Phi:\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$

is a morphism of $C^{\infty}(M)$ -modules. If in addition $T$ is a differential operator, then $T^{\Phi}$ is also a differential operator of the same order.
(2)

If $T:\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))\to\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$ is a morphism of $C^{\infty}(M)$ -modules that respects the gradation, then $T^{\Phi}:\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ respects the filtration and we have

${\rm gr}(T^{\Phi})=T.$
(3)

If $D:\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ respects the filtration, then $D-({\rm gr}(D))^{\Phi}$ strictly increases weights, and so

${\rm gr}(D)={\rm gr}\left(({\rm gr}(D))^{\Phi}\right).$

Proof.

These properties are easily checked on $\Theta^{I}$ and $\langle\Theta\rangle^{I}$ having fixed a local orthonormal adapted coframe $\Theta$ . ∎

2.5.2. Scalar products on $\wedge^{\bullet}T_{x}^{}M$ and of $\wedge^{\bullet}{\rm gr}(T_{x}^{}M)$

For each $x\in M$ , the scalar products $g_{T_{x}M}$ and $g_{{\rm gr}(T_{x}M)}$ on $T_{x}M$ and ${\rm gr}(T_{x}M)$ respectively induce scalar products on their duals $T_{x}^{*}M$ and ${\rm gr}(T_{x}M)^{*}\cong{\rm gr}(T_{x}^{\ast}M)$ , and then scalar products $g_{\wedge^{\bullet}T_{x}^{*}M}$ on $\wedge^{\bullet}T_{x}^{*}M$ and $g_{\wedge^{\bullet}{\rm gr}(T_{x}^{*}M)}$ on $\wedge^{\bullet}{\rm gr}(T_{x}^{*}M)$ via the formula in (2.15) below and the orthogonal decompositions:

\wedge^{\bullet}T_{x}^{*}M=\oplus^{\perp}\wedge^{k}T_{x}^{*}M\quad\mbox{and}\quad\wedge^{\bullet}{\rm gr}(T_{x}^{*}M)=\oplus^{\perp}\wedge^{k}{\rm gr}(T_{x}^{*}M)\,.

We will denote all these scalar product as $\langle\cdot,\cdot\rangle$ , their meaning being clear from the context or from an indexation. We have

(2.15)

\langle\alpha_{1}\wedge\ldots\wedge\alpha_{k},\beta_{1}\wedge\ldots\wedge\beta_{k}\rangle_{\wedge^{k}T_{x}^{*}M}:=\det\left(\langle\alpha_{i},\beta_{j}\rangle_{T_{x}^{*}M}\right)_{1\leq i,j\leq k},

where $\alpha_{1},\ldots,\alpha_{k},\beta_{1}\ldots,\beta_{k}\in T_{x}^{*}M$ , and similarly for ${\rm gr}(\wedge^{k}T_{x}^{*}M)\cong\wedge^{k}{\rm gr}(T_{x}^{\ast}M)$ .

We check readily that the map $\Phi_{x}$ defined above sends the scalar product of $\wedge^{\bullet}T_{x}^{*}M$ onto the one of $\wedge^{\bullet}{\rm gr}(T_{x}^{*}M)$ :

(2.16)

\Phi_{x}\colon\wedge^{\bullet}T_{x}^{\ast}M\to\wedge^{\bullet}{\rm gr}(T_{x}^{\ast}M)\ ,\quad\langle\Phi_{x}(\alpha),\Phi_{x}(\beta)\rangle=\langle\alpha,\beta\rangle\ \forall\,\alpha,\beta\in\wedge^{\bullet}T_{x}^{*}M\,.

Given an orthonormal local frame $\mathbb{X}$ or equivalently an orthonormal local coframe $\Theta$ , we check readily that the bases $\Theta^{k}(x)=(\Theta^{I})_{I}(x)$ and $\langle\Theta^{k}\rangle(x)=(\langle\Theta^{I}\rangle)_{I}(x)$ of $\wedge^{k}T_{x}^{*}M$ and of $\wedge^{k}{\rm gr}(T_{x}^{*}M)$ respectively are orthonormal. We will use the following notation to denote the elements of the bases in degree 0 and $n$ :

1=\Theta^{\emptyset}\text{ for }\wedge^{0}T^{\ast}_{x}M\ \text{ and }{\rm vol}_{\Theta}(x):=\theta^{1}(x)\wedge\ldots\wedge\theta^{n}(x)\text{ for }\wedge^{n}T_{x}^{\ast}M

and

1=\langle\Theta\rangle^{\emptyset}\text{ for }\wedge^{0}{\rm gr}(T_{x}^{\ast}M)\text{ and }\langle{\rm vol}_{\Theta}\rangle(x):=\langle\theta^{1}\rangle(x)\wedge\ldots\wedge\langle\theta^{n}\rangle(x)\text{ for }\wedge^{n}{\rm gr}(T_{x}^{\ast}M)\,.

With these scalar products, we can now define transposes and obtain some of their properties: if $T$ and $D$ are algebraic maps acting linearly on $\wedge^{\bullet}{\rm gr}(T^{*}M)$ and $\Omega^{\bullet}(M)$ respectively, then we define their transpose $T^{t}$ and $D^{t}$ as the algebraic maps acting linearly on $\wedge^{\bullet}{\rm gr}(T^{*}M)$ and $\Omega^{\bullet}(M)$ by their pointwise transpose, that is, $(T^{t})_{x}$ and $(D^{t})_{x}$ are the transpose of $T_{x}$ and $D_{x}$ for the scalar products $g_{\wedge^{\bullet}{\rm gr}(T_{x}^{*}M)}$ and $g_{\wedge^{\bullet}T_{x}^{*}M}$ at each point $x\in M$ . We check readily that if $T$ is an algebraic map acting linearly on $\wedge^{\bullet}{\rm gr}(T^{*}M)$ respecting the gradation, then so is $T^{t}$ . Moreover, the map $D:=\Phi^{-1}\circ T\circ\Phi$ and its transpose $D^{t}$ are both algebraic map acting linearly on $\Omega^{\bullet}(M)$ respecting the gradation, and satisfy:

D^{t}=\Phi^{-1}\circ T^{t}\circ\Phi\qquad\mbox{and}\qquad{\rm gr}(D^{t})=({\rm gr}(D))^{t}.

2.5.3. Hodge-star, scalar product on $\Omega_{c}^{\bullet}(M)$ and $\Gamma_{c}(\wedge^{\bullet}{\rm gr}(T^{*}M))$ , and transpose

The Hodge star operator on $\wedge^{\bullet}T_{x}^{*}M$ or $\wedge^{\bullet}{\rm gr}(T_{x}^{*}M)$ is the map defined via

\star:\wedge^{k}T_{x}^{*}M\overset{\cong}{\longrightarrow}\wedge^{n-k}T_{x}^{*}M\,,\ \alpha\wedge\star\beta:=\langle\alpha,\beta\rangle{\rm vol}_{\Theta}(x)\ \forall\,\alpha,\beta\in\wedge^{k}T_{x}^{*}M,\ k=1,\ldots,n\,,

and similarly on $\wedge^{\bullet}{\rm gr}(T_{x}^{*}M)$ . As $\Phi_{x}$ maps the scalar product of $\wedge^{\bullet}T_{x}^{*}M$ to the one of $\wedge^{\bullet}{\rm gr}(T_{x}^{*}M)$ , (see (2.16)), it also maps the Hodge star operator of $\wedge^{\bullet}T_{x}^{*}M$ to the one of $\wedge^{\bullet}{\rm gr}(T_{x}^{*}M)$ :

(2.17)

\forall\alpha,\beta\in\wedge^{\bullet}T_{x}^{*}M\qquad\star\Phi_{x}(\alpha)=\Phi_{x}(\star\alpha).

The $\star$ operator naturally extends to an algebraic operator on $\Omega^{\bullet}(M)$ and on $\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$ respectively. Moreover, from the definition of $\star$ , we have that

\alpha\wedge\star\beta=\beta\wedge\star\alpha\quad\mbox{on}\ \Omega^{\bullet}(M)\ \mbox{and}\ \Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M)),

and

\star 1={\rm vol}_{\Theta},\ \star{\rm vol}_{\Theta}=1,\quad\mbox{and}\quad\star 1=\langle{\rm vol}_{\Theta}\rangle,\ \star\langle{\rm vol}_{\Theta}\rangle=1.

Moreover, if a local orthonormal adapted coframe $\Theta$ is fixed, the Hodge star operators on $\Theta^{I}$ and $\langle\Theta\rangle^{I}$ with multi-index $I=(i_{1},\ldots,i_{k})$ are given by

\star(\Theta^{I})=(-1)^{\sigma(I)}\Theta^{\bar{I}},\qquad\star(\langle\Theta\rangle^{I})=(-1)^{\sigma(I)}\langle\Theta\rangle^{\bar{I}}

where $\bar{I}=(\bar{i}_{1},\ldots,\bar{i}_{n-k})$ with $\bar{i}_{1}<\bar{i}_{2}<\ldots<\bar{i}_{n-k}$ is the multi-index complementing $I$ , and $(-1)^{\sigma(I)}$ is the sign of the permutation $\sigma(I)=i_{1}\cdots i_{k}\,\bar{i}_{1}\cdots\bar{i}_{n-k}$ . This completely characterises the Hodge star operators on both $\Omega^{\bullet}(M)$ and on $\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$ .

The previous properties imply readily that for any $\alpha,\beta$ in either $\Omega^{k}(M)$ or $\Gamma(\wedge^{k}{\rm gr}(T^{*}M))$ , we have

\star\star\beta=(-1)^{k(n-k)}\beta\qquad\mbox{and}\qquad\langle\star\alpha,\star\beta\rangle=\langle\alpha,\beta\rangle\,.

We can now define the so-called $L^{2}$ -inner products on the subspaces $\Omega^{\bullet}_{c}(M)$ and $\Gamma_{c}(\wedge^{\bullet}{\rm gr}(T^{*}M))$ of forms with compact support in $\Omega^{\bullet}(M)$ and $\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$ respectively via

\langle\alpha,\beta\rangle_{L^{2}}:=\int_{M}\alpha\wedge\star\beta=\int_{M}\langle\alpha,\beta\rangle\,{\rm vol}\ ,\ \forall\,\alpha,\beta\in\Omega^{\bullet}_{c}(M)\text{ or }\Gamma_{c}(\wedge^{\bullet}{\rm gr}(T^{*}M))\,.

We have $\Phi(\Omega^{\bullet}_{c}(M))=\Gamma_{c}(\wedge^{\bullet}{\rm gr}(T^{*}M))$ and

(2.18)

\forall\alpha,\beta\in\Omega^{\bullet}_{c}(M)\qquad\langle\alpha,\beta\rangle_{L^{2}}=\langle\Phi(\alpha),\Phi(\beta)\rangle_{L^{2}}.

2.5.4. Properties of the transpose for the $L^{2}$ -inner product

The above definitions of the Hodge operator and of the scalar product show that if $D$ is a differential operator on either $\Omega^{\bullet}(M)$ or $\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$ , then so are $\star D$ and $D\star$ , as well as the formal transpose $D^{t}$ defined as $\langle D\alpha,\beta\rangle_{L^{2}}=\langle\alpha,D^{t}\beta\rangle_{L^{2}}$ for any $\alpha\in\Omega_{c}^{k}(M)$ or $\Gamma_{c}(\wedge^{k}{\rm gr}(T^{\ast}M))$ and any $\beta\in\Omega^{k+1}_{c}(M)$ or $\Gamma_{c}(\wedge^{k+1}{\rm gr}(T^{\ast}M))$ .

In the particular case of the de Rham differential $d$ on forms, we observe that the transpose of $d^{(k)}=d:\Omega^{k}(M)\to\Omega^{k+1}(M)$ is a differential operator that satisfies

d^{(k,t)}=(-1)^{kn+1}\star d^{(n-k-1)}\star\colon\Omega^{k+1}(M)\to\Omega^{k}(M)\,.

We can also consider the transpose of the algebraic linear map $\tilde{d}^{(k)}=\tilde{d}:\Omega^{k}(M)\to\Omega^{k+1}(M)$ of Subsection 2.2.1. The linear map $\tilde{d}^{t}$ is then algebraic and satisfies

\tilde{d}^{(k,t)}=(-1)^{kn+1}\star\tilde{d}^{(n-k-1)}\star\colon\Omega^{k+1}(M)\to\Omega^{k}(M)\,.

It is important to realise that the transpose may not respect the filtration in the following sense: for a general differential operator $D\colon\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ , it is true that the transpose $D^{t}$ is a differential operator, but $D$ respecting the filtration (2.11) will not imply always that $D^{t}$ does so. This may not be true even for $d$ and $\tilde{d}$ , and two explicit ‘counter-examples’ of this situation are given in Appendix B. However, the following lemma guarantees that $D^{t}$ will also respect the filtration when $D$ is of the form $D=T^{\Phi}=\Phi^{-1}\circ T\circ\Phi$ with $T$ a homomorphism of the $C^{\infty}(M)$ -module $\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$ :

Lemma 2.20.

We continue with the setting of Lemma 2.19, where $T:\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))\to\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$ is a morphism of $C^{\infty}(M)$ -modules. We consider the map $D:=T^{\Phi}=\Phi^{-1}\circ T\circ\Phi$ acting on $\Omega^{\bullet}(M)$ . Let us further assume that $T$ respects the gradation.

(1)

The $C^{\infty}(M)$ -module morphism $\star T\star\colon\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))\to\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$ also respects the gradation, and $(\star T\star)^{\Phi}=\star T^{\Phi}\star.$ Consequently, the $C^{\infty}(M)$ -module morphism $D:=T^{\Phi}:\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ respects the filtration and we have ${\rm gr}(\star D\star)=\star{\rm gr}(D)\star$ .
(2)

If in addition, $T$ is a differential operator, then $T^{t}$ is also a differential operator of the same order as $T$ , it respects the gradation, and satisfies $(T^{t})^{\Phi}=(T^{\Phi})^{t}$ . Consequently, the differential operators on smooth forms $D:=T^{\Phi}$ and its transpose $D^{t}$ respect the filtration, and ${\rm gr}(D^{t})=({\rm gr}(D))^{t}.$

Proof.

The equality $(\star T\star)^{\Phi}=\star T^{\Phi}\star$ follows from (2.17).

Let $\Theta$ be a local adapted frame. We observe that $\langle\Theta\rangle^{I}$ is of weight $|I|$ while $\star\langle\Theta\rangle^{I}=(-1)^{\sigma(I)}\langle\Theta\rangle^{\bar{I}}$ is of weight $|\bar{I}|=Q-|I|$ where $Q:=\upsilon_{1}+\ldots+\upsilon_{n}$ is the weight of the volume form.

Since $T$ respects the gradation and the weight of $\langle\Theta\rangle^{\bar{I}}$ is $|\bar{I}|$ , $T(\langle\Theta\rangle^{\bar{I}})$ has the same weight $|\bar{I}|$ . Consequently, $T(\langle\Theta\rangle^{\bar{I}})$ is a $C^{\infty}(M)$ -linear combination of basis elements $\langle\Theta\rangle^{J}$ where the multi-indices $J$ have weight $|J|=|\bar{I}|=Q-|I|$ . Finally, $\star T(\Theta^{\bar{I}})$ has weight $Q-|J|=|I|$ . We have therefore shown that the morphism $\star T\star$ maps elements of weight $|I|$ to elements of weight $|I|$ , and so $\star T\star$ respects the gradation. Lemma 2.19 implies the rest of Part (1).

Part (2) follows from the definition of $D^{t}$ and its link with $\star$ covered in Subsection 2.5.3. ∎

2.5.5. Properties of ${\rm gr}$ for algebraic maps

Here, we describe some properties of ${\rm gr}$ in relation with algebraic maps (for the latter concept see Section 2.3.2).

The following properties are easily checked.

Lemma 2.21.

Let $M$ be a smooth manifold whose tangent bundle is filtered by vector subbundles as in (2.9).

(1)

Any algebraic morphism of $C^{\infty}(M)$ -module $D$ acting either on $\Omega^{\bullet}(M)$ or $\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$ is determined by its pointwise restriction $D_{x}$ at each $x\in M$ . Conversely, any smooth map on $\wedge^{\bullet}T^{*}M$ or $\wedge^{\bullet}{\rm gr}(T^{*}M)$ that is linear at each fibre yields an algebraic morphism of $C^{\infty}(M)$ -modules.
(2)

If a morphism of $C^{\infty}(M)$ -modules $D$ acting on $\Omega^{\bullet}(M)$ is algebraic and respects the filtration, then ${\rm gr}(D)$ is a morphism of $C^{\infty}(M)$ -module acting on $\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$ which is also an algebraic map that respects the gradation. Moreover, if $D$ is invertible, then ${\rm gr}(D)$ is also invertible, and $D^{-1}$ and ${\rm gr}(D)^{-1}$ are algebraic.

The isomorphism $\Phi$ together with the concept of algebraic maps allow us to construct inverses in some cases, and this provides a partial converse to (2) in Lemma 2.21 as follows.

Proposition 2.22.

Let $M$ be a smooth manifold whose tangent bundle is filtered by vector subbundles as in (2.9). Assume that $M$ is equipped with a metric $g_{TM}$ . We consider the corresponding map $\Phi$ defined in Section 2.5.1.

Let $D:\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ be a morphism of $C^{\infty}(M)$ -modules that respects the filtration and such that ${\rm gr}(D)$ is algebraic. Assume also that ${\rm gr}(D)$ is fiberwise invertible in the sense that its restriction ${\rm gr}(D)_{x}$ at each $x\in M$ is a linear invertible map on $\wedge^{\bullet}T_{x}^{*}M$ . Then the map $D$ is invertible and the map $D^{-1}:\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ respects the filtration. Moreover, ${\rm gr}(D^{-1})={\rm gr}(D)^{-1}$ , and if $D$ is in addition a differential operator acting on $\Omega^{\bullet}(M)$ , then so is its inverse.

Proof.

By (2) in Lemma 2.19, the map $T:=({\rm gr}(D))^{\Phi}\colon\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ respects the filtration and ${\rm gr}(T)^{-1}={\rm gr}(D)^{-1}$ . Moreover, by (2) of Lemma 2.21, $T$ is invertible and its inverse is $T^{-1}=({\rm gr}(D)^{-1})^{\Phi}$ .

By Lemma 2.16, we have

{\rm gr}(T^{-1}D)={\rm gr}(T^{-1}){\rm gr}(D)={\rm gr}(D)^{-1}{\rm gr}(D)=\text{\rm I}\ ,\mbox{ so }{\rm gr}(T^{-1}D-\text{\rm I})={\rm gr}(T^{-1}D)-\text{\rm I}=0\ .

Hence $T^{-1}D-\text{\rm I}=:B$ is nilpotent with $B^{N_{0}}=0$ by (2.14). Since $T^{-1}D=\text{\rm I}+B$ and $\text{\rm I}+B$ is invertible with $(\text{\rm I}+B)^{-1}=\sum_{j=0}^{N_{0}}(-1)^{j}B^{j}$ , we have $(\text{\rm I}+B)^{-1}T^{-1}D=\text{\rm I}$ . This provides a left inverse for $D$ and the formula below with $B=B_{L}$ . We can do the same on the right. ∎

Corollary 2.23.

We continue with the setting of Proposition 2.22 and its proof. The following formulae holds for $D^{-1}:\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ :

D^{-1}=\sum_{j=0}^{N_{0}}(-1)^{j}B_{L}^{j}T^{-1}=T^{-1}\sum_{j=0}^{N_{0}}(-1)^{j}B_{R}^{j}

where $T={\rm gr}(D)^{\Phi}=\Phi{\rm gr}(D)^{-1}\Phi^{-1}$ , and $B_{L}=T^{-1}D-\text{\rm I}$ while $B_{R}=DT^{-1}-\text{\rm I}$ .

3. Filtered manifolds and osculating objects

In this section, we discuss the setting of filtered manifolds and some complexes naturally associated with them. In particular, we will define the osculating differential $d_{\mathfrak{g}M}$ and study its relation to the de Rham complex $(\Omega^{\bullet}(M),d)$ . We will also use the algebraic part $\tilde{d}$ of $d$ . The main result of this section is in Proposition 3.3 stating that ${\rm gr}(d)={\rm gr}(\tilde{d})=d_{\mathfrak{g}M}$ .

3.1. Setting and definitions

A filtered manifold is a smooth manifold $M$ equipped with a filtration of the tangent bundle $TM$ by vector subbundles

M\times\{0\}=H^{0}\subseteq H^{1}\subseteq\ldots\subseteq H^{s}=TM

satisfying

(3.1)

[\Gamma(H^{i}),\Gamma(H^{j})]\subseteq\Gamma(H^{i+j})\,,

with the convention that $H^{i}=TM$ when $i>s$ .

Example 3.1.

Any contact manifold, or more generally any regular subRiemannian manifold, is a filtered manifold, see Section 4.6.1. A related class of examples is given by nilpotent Lie groups equipped with a left-invariant filtration [LDT22].

3.1.1. Bundle of osculating Lie groups and algebras

For each $x\in M$ , the quotient ${\rm gr}(T_{x}M)=\oplus_{i=1}^{s_{0}}\left(H^{i}_{x}/H^{i-1}_{x}\right)$ is naturally equipped with a Lie bracket $[\cdot,\cdot]_{\mathfrak{g}_{x}M}$ , since

\forall\,f,g\in C^{\infty}(M),\ X\in\Gamma(H^{i}),\ Y\in\Gamma(H^{j})\qquad[fX,gY]=fg[X,Y]+\Gamma(H^{i+j-1}).

When ${\rm gr}(T_{x}M)$ is equipped with this Lie bracket, we denote the resulting Lie algebra as

\mathfrak{g}_{x}M:=({\rm gr}(T_{x}M),[\cdot,\cdot]_{\mathfrak{g}_{x}M})\,.

It is naturally graded by

\mathfrak{g}_{x}M=\oplus_{i=1}^{s_{0}}\left(H^{i}_{x}/H^{i-1}_{x}\right),

and is therefore nilpotent. We denote by $G_{x}M$ the corresponding connected simply connected nilpotent Lie group (sometimes called the nilpotentisation or the tangent cone [Mit85] of $M$ at $x$ ). The unions

GM:=\cup_{x\in M}G_{x}M\qquad\mbox{and}\qquad\mathfrak{g}M:=\cup_{x\in M}\mathfrak{g}_{x}M,

are naturally equipped with a smooth bundle structure that are called [RS76, vEY17] the bundles of osculating Lie groups and Lie algebras over $M$ .

We observe that the notions of weights defined in Section 2.4.3 and for the graded Lie algebra $\mathfrak{g}_{x}M$ coincide. We therefore obtain an analogous decomposition of $\wedge^{\bullet}\mathfrak{g}_{x}^{*}M$ in terms of weights and degrees:

(3.2)

\wedge^{\bullet}\mathfrak{g}_{x}^{*}M=\oplus_{k,w\in\mathbb{N}_{0}}\wedge^{k,w}\mathfrak{g}_{x}^{*}M\ ,\quad\forall\,x\in M\ .

We introduce the following vocabulary.

Definition 3.2.

The elements of $G\Omega^{\bullet}(M):=\Gamma(\wedge^{\bullet}\mathfrak{g}^{\ast}M)$ are called osculating forms. Moreover, for any $k,w\in\mathbb{N}_{0}$ ,

G\Omega^{k}(M):=\Gamma(\wedge^{k}\mathfrak{g}^{*}M)\qquad\mbox{and}\qquad G\Omega^{k,w}(M):=\Gamma(\wedge^{k,w}\mathfrak{g}^{*}M)

are called osculating $k$ -forms (or osculating forms of degree $k$ ) and osculating $k$ -forms of weights $w$ respectively.

We say that a map $T\colon G\Omega^{\bullet}(M)\to G\Omega^{\bullet}(M)$ respects the weights of osculating forms when $T(G\Omega^{\bullet,w}(M))\subseteq G\Omega^{\bullet,w}(M)$ , and it increases the weights when $T(G\Omega^{\bullet,w}(M))\subseteq\oplus_{w^{\prime}\geq w}G\Omega^{\bullet,w^{\prime}}(M)$ .

3.1.2. The osculating Chevalley-Eilenberg differential

For each $x\in M$ , $d_{\mathfrak{g}_{x}M}$ denotes the Chevalley-Eilenberg differential on the Lie group $G_{x}M$ viewed as the map

d_{\mathfrak{g}_{x}M}:\wedge^{\bullet}\mathfrak{g}_{x}^{*}M\to\wedge^{\bullet}\mathfrak{g}_{x}^{*}M

defined via $d_{\mathfrak{g}_{x}M}(\wedge^{0}\mathfrak{g}_{x}^{*}M)=\{0\}$ for $k=0$ , for $k=1$

(3.3)

\forall\,\alpha\in\wedge^{1}\mathfrak{g}_{x}^{*}M\,,\ V_{0},V_{1}\in\mathfrak{g}_{x}M\ ,\quad d_{\mathfrak{g}_{x}M}\alpha(V_{0},V_{1})=-\alpha([V_{0},V_{1}]_{\mathfrak{g}_{x}M}),

and more generally for $k>0$ , for any $\alpha\in\wedge^{k}\mathfrak{g}_{x}^{*}M$ and $V_{0},\ldots V_{k}\in\mathfrak{g}_{x}M$ ,

(3.4)

d_{\mathfrak{g}_{x}M}\alpha(V_{0},\ldots,V_{k})=\sum_{0\leq i<j\leq k}(-1)^{i+j}\alpha([V_{i},V_{j}]_{\mathfrak{g}_{x}M},V_{0},\ldots,\hat{V}_{i},\ldots,\hat{V}_{j},\ldots,V_{k}).

It is well known [FT23] that $d_{\mathfrak{g}_{x}M}$ is a linear map such that

d_{\mathfrak{g}_{x}M}^{2}=0\ ,\quad d_{\mathfrak{g}_{x}M}(\wedge^{k}\mathfrak{g}_{x}^{*}M)\subseteq\wedge^{k+1}\mathfrak{g}_{x}^{*}M,

and that it satisfies the Leibniz property on $\wedge^{\bullet}\mathfrak{g}_{x}^{*}M$ , i.e.

\forall\,\alpha\in\wedge^{k}\mathfrak{g}_{x}^{*}M,\ \beta\in\wedge^{\bullet}\mathfrak{g}_{x}^{*}M\ ,\quad d_{\mathfrak{g}_{x}M}(\alpha\wedge\beta)=(d_{\mathfrak{g}_{x}M}\alpha)\wedge\beta+(-1)^{k}\alpha\wedge(d_{\mathfrak{g}_{x}M}\beta).

Moreover, $d_{\mathfrak{g}_{x}M}$ also preserves the weights of $\wedge^{\bullet}\mathfrak{g}_{x}^{*}M$ :

d_{\mathfrak{g}_{x}M}(\wedge^{k,w}\mathfrak{g}_{x}^{*}M)\subseteq\wedge^{k+1,w}\mathfrak{g}_{x}^{*}M.

By construction, the algebraic map $d_{\mathfrak{g}M}$ defined by

d_{\mathfrak{g}M}:G\Omega^{\bullet}(M)\to G\Omega^{\bullet}(M)\qquad(d_{\mathfrak{g}M})_{x}=d_{\mathfrak{g}_{x}M},\ \forall\,x\in M,

is smooth. We call $(G\Omega^{\bullet}(M),d_{\mathfrak{g}M})$ the osculating complex or osculating Chevalley-Eilenberg differential.

3.1.3. The osculating box operator

Here we assume that the vector bundle $\wedge^{\bullet}\mathfrak{g}^{*}M$ is equipped with a metric $g_{\wedge^{\bullet}\mathfrak{g}^{*}M}$ : each $\wedge^{\bullet}\mathfrak{g}_{x}^{*}M$ is equipped with a scalar product $g_{\wedge^{\bullet}\mathfrak{g}_{x}^{*}M}$ with a smooth dependence in $x\in M$ . This scalar product allows us to consider, at each point $x\in M$ , the transpose of the osculating differential $d_{\mathfrak{g}M}$ , and then to define

\Box_{\mathfrak{g}M}:=d_{\mathfrak{g}M}d_{\mathfrak{g}M}^{t}+d_{\mathfrak{g}M}^{t}d_{\mathfrak{g}M}\,.

We call this operator the osculating box. As $d_{\mathfrak{g}_{M}}$ is algebraic, so are $d_{\mathfrak{g}M}^{t}$ and $\Box_{\mathfrak{g}M}$ .

Let us now assume that the decomposition (3.2) is orthogonal for the scalar product $g_{\wedge^{\bullet}\mathfrak{g}_{x}^{*}M}$ for each $x\in M$ . This arises naturally when the manifold is filtered and Riemannian (see Section 2.5). With this assumption, we readily check that $d_{\mathfrak{g}M}^{t}$ preserves the weights, i.e. $d_{\mathfrak{g}_{x}M}^{t}(\wedge^{k,w}\mathfrak{g}_{x}^{*}M)\subseteq\wedge^{k-1,w}\mathfrak{g}_{x}^{*}M$ , and $\Box_{\mathfrak{g}M}$ respects the degree and weights of osculating forms:

\Box_{\mathfrak{g}_{x}M}(\wedge^{k,w}\mathfrak{g}_{x}^{*}M)\subseteq\wedge^{k,w}\mathfrak{g}_{x}^{*}M.

Moreover, given the scalar product $g_{\wedge^{\bullet}\mathfrak{g}_{x}^{*}M}$ , the linear map

\Box_{\mathfrak{g}_{x}M}:=(\Box_{\mathfrak{g}M})_{x}=d_{\mathfrak{g}_{x}M}d_{\mathfrak{g}_{x}M}^{t}+d_{\mathfrak{g}_{x}M}^{t}d_{\mathfrak{g}_{x}M}\colon\wedge^{\bullet}\mathfrak{g}_{x}^{*}M\to\wedge^{\bullet}\mathfrak{g}_{x}^{*}M

is symmetric. We denote by $\Pi_{\mathfrak{g}_{x}M}$ the spectral orthogonal projection onto its kernel

E_{\mathfrak{g}_{x}M}:=\ker(\Box_{\mathfrak{g}_{x}M})=\ker d_{\mathfrak{g}_{x}M}\cap\ker d_{\mathfrak{g}_{x}M}^{t}={\rm Im}\,\Pi_{\mathfrak{g}_{x}M}.

Note that the image of $\Box_{\mathfrak{g}_{x}M}$ is

F_{\mathfrak{g}_{x}M}:={\rm Im}\,(\Box_{\mathfrak{g}_{x}M})={\rm Im}\,d_{\mathfrak{g}_{x}M}+{\rm Im}\,d_{\mathfrak{g}_{x}M}^{t}=\ker\Pi_{\mathfrak{g}_{x}M}=E_{\mathfrak{g}_{x}M}^{\perp}.

By [FT23, Lemma 2.1], the complex $(F_{\mathfrak{g}M}^{\bullet},d_{\mathfrak{g}M})$ is acyclic, i.e. $d_{\mathfrak{g}M}(F_{\mathfrak{g}M})\subseteq F_{\mathfrak{g}M}$ , and the resulting cohomology is trivial:

d_{\mathfrak{g}_{x}M}(F_{\mathfrak{g}_{x}M})=\ker(d_{\mathfrak{g}_{x}M}:F_{\mathfrak{g}_{x}M}\to F_{\mathfrak{g}_{x}M})\subseteq\wedge^{\bullet}\mathfrak{g}_{x}^{*}M.

For $|z|=\varepsilon$ small enough, $z-\Box_{\mathfrak{g}_{x}M}$ is invertible, and by the Cauchy residue formula we have

\Pi_{\mathfrak{g}_{x}M}=\frac{1}{2\pi i}\oint_{|z|=\varepsilon}(z-\Box_{\mathfrak{g}_{x}M})^{-1}dz,

where the contour integration is over a circle about 0 of radius $\varepsilon>0$ small enough. This defines a smooth algebraic map $\Pi_{\mathfrak{g}M}$ acting on $G\Omega^{\bullet}(M)$ that respects degrees and weights.

3.1.4. The partial inverse $d^{-1}_{\mathfrak{g}M}$

Proceeding as in [FT23], at each $x\in M$ , a partial inverse $d_{\mathfrak{g}_{x}M}^{-1}$ of $d_{\mathfrak{g}_{x}M}$ can be defined as

d_{\mathfrak{g}_{x}M}^{-1}:=(d_{\mathfrak{g}_{x}M})^{-1}{\rm pr}_{{\rm Im}\,d_{\mathfrak{g}_{x}M}}\,,

where we write ${\rm pr}_{S}$ to denote the orthogonal projection onto any $S$ closed subspace of $\wedge^{\bullet}\mathfrak{g}_{x}^{*}M$ .

Note that in our construction in Section 4, we will not use $d_{\mathfrak{g}_{x}M}^{-1}$ . However, it is used in Rumin’s construction, see Section 4.5.

where we show that the two constructions coincide.

The resulting map $d_{\mathfrak{g}M}^{-1}$ on $G\Omega(M)$ is a smooth map that respects the filtration, and such that

(d_{\mathfrak{g}M}^{-1})^{2}=0\ \text{ and }\,d_{\mathfrak{g}_{x}M}^{-1}(\wedge^{k+1,w}\mathfrak{g}_{x}^{*}M)\subseteq\wedge^{k,w}\mathfrak{g}_{x}^{*}M\ ,\ \forall\,k,w\in\mathbb{N}_{0}\,.

By [FT23, Section 2], the kernel and image of $d_{\mathfrak{g}M}^{-1}$ coincide with the one of $d_{\mathfrak{g}M}^{t}$ :

\ker d_{\mathfrak{g}_{x}M}^{t}=\ker d_{\mathfrak{g}_{x}M}^{-1}\quad\mbox{and}\quad{\rm Im}\,d_{\mathfrak{g}_{x}M}^{t}={\rm Im}\,d_{\mathfrak{g}_{x}M}^{-1},

E_{\mathfrak{g}_{x}M}=\ker d_{\mathfrak{g}_{x}M}\cap\ker d_{\mathfrak{g}_{x}M}^{-1}\quad\mbox{and}\quad F_{\mathfrak{g}_{x}M}={\rm Im}\,d_{\mathfrak{g}_{x}M}+{\rm Im}\,d_{\mathfrak{g}_{x}M}^{-1}.

By [FT23, Proposition 2.2],

\Pi_{\mathfrak{g}M}=\text{\rm I}-d_{\mathfrak{g}M}^{-1}d_{\mathfrak{g}M}-d_{\mathfrak{g}M}d_{\mathfrak{g}M}^{-1}

and we have

d_{\mathfrak{g}M}^{-1}\Pi_{\mathfrak{g}M}=\Pi_{\mathfrak{g}M}d_{\mathfrak{g}M}^{-1}=0\ ,\quad d_{\mathfrak{g}M}\Pi_{\mathfrak{g}M}=\Pi_{\mathfrak{g}M}d_{\mathfrak{g}M}=0\,.

3.2. The maps $d$ and $\tilde{d}$ on a filtered manifold

Let us show that ${\rm gr}(d)$ and ${\rm gr}(\tilde{d})$ are well-defined, crucial operators, and that both $(\Gamma(\wedge^{\bullet}{\rm gr}(T^{\ast}M)),{\rm gr}(d))$ and $(\Gamma(\wedge^{\bullet}{\rm gr}(T^{\ast}M)),{\rm gr}(\tilde{d}))$ coincide with the osculating complex.

Proposition 3.3.

The maps $d$ and $\tilde{d}$ respect the filtration with

(d-\tilde{d})\Omega^{k,\geq w}(M)\subseteq\Omega^{k,\geq w+1}(M),

for any degree $k\in\mathbb{N}_{0}$ and weight $w\in\mathbb{N}_{0}$ . Moreover, the maps ${\rm gr}(d)$ and ${\rm gr}(\tilde{d})$ acting on $G\Omega^{\bullet}(M)\cong\Gamma(\wedge^{\bullet}{\rm gr}(T^{\ast}M))$ coincide with $d_{\mathfrak{g}M}$ :

{\rm gr}(d)={\rm gr}(\tilde{d})=d_{\mathfrak{g}M}\,.

Proof.

From the definitions of $d$ and $\tilde{d}$ , both differentials as maps $\Omega^{1}(M)\to\Omega^{2}(M)$ respect the corresponding filtrations. As they obey the Leibniz rule, both $d$ and $\tilde{d}$ respect the filtration on $\Omega^{\bullet}(M)$ . Moreover, ${\rm gr}(d)$ and ${\rm gr}(\tilde{d})$ satisfy the Leibniz property on $G\Omega^{\bullet}(M)\cong\Gamma(\wedge^{\bullet}{\rm gr}(T^{\ast}M))$ .

Let $\omega\in\Omega^{k,\geq w}(M)$ and for each $j=0,\ldots,k$ take $V_{j}\in H^{w_{i(j)}}$ with $w_{i(0)}+\ldots+w_{i(k)}\leq w$ . Since $w_{i(0)}+\ldots+\hat{w}_{i(l)}+\ldots+w_{i(k)}<w$ , we have that $\omega(V_{0},\ldots,\hat{V}_{l},\ldots,V_{k})=0$ , and so $(d-\tilde{d})\omega(V_{0},\ldots,V_{k})=0$ by (2.8).

This shows $(d-\tilde{d})\Omega^{k,\geq w}(M)\subseteq\Omega^{k,\geq w+1}(M)$ , and implies ${\rm gr}(d)={\rm gr}(\tilde{d})$ . We are left to prove that ${\rm gr}(\tilde{d})=d_{\mathfrak{g}M}$ .

If $\omega\in\Omega^{1}(M)$ and $V_{i_{1}}\in\Gamma(H^{w_{i(1)}}),V_{i_{2}}\in\Gamma(H^{w_{i(2)}})$ , then by Lemma 2.3 and (3.1).

	$\displaystyle\tilde{d}\omega(V_{1}\,{\rm mod}\,H^{w_{i(1)}-1},V_{2}\,{\rm mod}\,H^{w_{i(2)}-1})$	$\displaystyle=-\omega([V_{1}\,{\rm mod}\,H^{w_{i(1)}-1},V_{2}\,{\rm mod}\,H^{w_{i(2)}-1}])$
		$\displaystyle=-\omega([V_{1},V_{2}]\,{\rm mod}\,H^{w_{i(1)}+w_{i(2)}-1})\,.$

Hence, we have that for any $\omega\in(H^{w_{j}})^{\perp}$ , $j=0,1,\ldots,s_{0}-1$

{\rm gr}(\tilde{d})(\omega\,{\rm mod}\,(H^{w_{j+1}})^{\perp})=-(\omega\,{\rm mod}\,(H^{w_{j+1}})^{\perp})([\cdot,\cdot]_{\mathfrak{g}M}).

We recognise $d_{\mathfrak{g}M}(\omega\,{\rm mod}\,(H^{w_{j+1}})^{\perp})$ from (3.3). Hence the maps $d_{\mathfrak{g}_{M}}$ and ${\rm gr}(\tilde{d})$ coincide on $G\Omega^{1}(M)$ . Since they both vanish on $G\Omega^{0}(M)$ and satisfy the Leibniz property, they coincide in fact on the whole $G\Omega^{\bullet}(M)$ . ∎

4. Construction of subcomplexes on Riemannian filtered manifolds

Here, we present the general scheme to construct subcomplexes on a filtered manifold $M$ whose tangent bundle is equipped with a metric $g_{TM}$ . It relies on the notions of a base differential and codifferential that we introduce in Section 4.1 (this terminology is borrowed from [GT53]). In Section 4.2, we construct certain operators $\Box,P,L$ which allow us to define the subcomplexes $(F_{0}^{\bullet},C)$ and $(E_{0}^{\bullet},D)$ , with the latter computing the same cohomology as de Rham’s (see Section 4.3). In Section 4.4, we also present an analogous construction, obtaining the operators $\tilde{\Box},\tilde{P},\tilde{L}$ and then the subcomplexes $(\tilde{F}_{0}^{\bullet},\tilde{C})$ and $(\tilde{E}_{0}^{\bullet},\tilde{D})$ , with the latter computing the same cohomology as the complex $(\Omega^{\bullet}(M),\tilde{d})$ . We present an alternative construction for $(E_{0}^{\bullet},D)$ and $(\tilde{E}_{0}^{\bullet},\tilde{D})$ in Section 4.5 that follows Rumin’s ideas.

4.1. Base differential and codifferential $d_{0}$ and $\delta_{0}$

Definition 4.1.

Let $d_{0}$ and $\delta_{0}$ be two differential algebraic complexes on a filtered manifold $M$ equipped with a metric $g_{TM}$ . We say that $(d_{0},\delta_{0})$ is a pair of base differential and codifferential when the following properties are satisfied:

•

$d_{0}$ increases the degree by one while $\delta_{0}$ decreases the degree by one, and both respect the filtration of $\Omega^{\bullet}(M)$ :

d_{0}(\Omega^{k,\geq w}(M))\subset\Omega^{k+1,\geq w}(M)\quad\mbox{and}\quad\delta_{0}(\Omega^{k+1,\geq w}(M))\subset\Omega^{k,\geq w}(M),

•

$d_{0}$ and $\delta_{0}$ are disjoint [Kos61], that is, for any $\alpha\in\Omega^{\bullet}(M)$

\displaystyle\delta_{0}d_{0}\alpha=0\Longrightarrow d_{0}\alpha=0\ \text{ and }\ d_{0}\delta_{0}\alpha=0\Longrightarrow\delta_{0}\alpha=0\,,

•

${\rm gr}(d_{0})=d_{\mathfrak{g}M}$ and ${\rm gr}(\delta_{0})=d_{\mathfrak{g}M}^{t}$ .

Above, the transpose is defined using a metric $g_{\wedge^{\bullet}\mathfrak{g}^{*}M}$ on the osculating bundle $\wedge^{\bullet}\mathfrak{g}^{*}M$ induced by $g_{TM}$ .

Given a pair $(d_{0},\delta_{0})$ of base differential and codifferential on a filtered manifold $M$ , we define the associated base box via

\Box_{0}:=d_{0}\,\delta_{0}+\delta_{0}\,d_{0}.

For any $x\in M$ , we define the operator

\Pi_{0,x}:=\frac{1}{2\pi i}\oint_{|z|=\epsilon}(z-\Box_{0,x})^{-1}dz,

for $\varepsilon>0$ small enough.

Proposition 4.2.

(1)

The operators $\Box_{0}$ and $\Pi_{0}$ are smooth and algebraic on $\Omega^{\bullet}(M)$ . They respect its filtration and keep the degrees of the forms constant:

\Box_{0}(\Omega^{k,\geq w}(M))\subset\Omega^{k,\geq w}(M)\quad\mbox{and}\quad\Pi_{0}(\Omega^{k,\geq w}(M))\subset\Omega^{k,\geq w}(M).

They also satisfy

{\rm gr}(\Box_{0})=\Box_{\mathfrak{g}M}\qquad\mbox{and}\qquad{\rm gr}(\Pi_{0})=\Pi_{\mathfrak{g}M}.

The algebraic subbundles

E_{0}:=\ker d_{0}\cap\ker\delta_{0}\quad\mbox{and}\quad F_{0}:={\rm Im}\,d_{0}+{\rm Im}\,\delta_{0},

admit subfiltrations and we have:

{\rm gr}(E_{0})=E_{\mathfrak{g}M}\qquad\mbox{and}\qquad{\rm gr}(F_{0})=F_{\mathfrak{g}M}.

(2)

We have

\Omega^{\bullet}(M)=E_{0}^{\bullet}\oplus F_{0}^{\bullet}\qquad\mbox{with}\qquad E_{0}=\ker\Box_{0}\qquad\mbox{and}\qquad F_{0}={\rm Im}\,\Box_{0}.

Moreover, $\Pi_{0}$ is the projection onto $E_{0}$ along $F_{0}$ .

Proof.

Part (1) is satisfied by construction. Disjointedness implies readily that $E_{0}\cap F_{0}=\{0\}$ . Since $E_{\mathfrak{g}M}\oplus F_{\mathfrak{g}M}=\wedge^{\bullet}\mathfrak{g}^{*}M$ , Part (1) implies that $\dim E_{0}+\dim F_{0}=\dim E_{\mathfrak{g}M}+\dim F_{\mathfrak{g}M}=\dim\Omega^{\bullet}(M)$ . Hence $E_{0}\oplus F_{0}=\Omega^{\bullet}(M)$ .

Disjointedness also implies $E_{0}=\ker\Box_{0}$ . Clearly, we also have ${\rm Im}\,\Box_{0}\subset F_{0}$ . From the Cauchy residue formula, proceeding as in the proof of [FT23, Proposition 4.1], we check that $\Pi_{0}^{2}=\Pi_{0}$ . In other words, $\Pi_{0}$ is a projection of $\Omega^{\bullet}(M)$ . We also check using the Cauchy residue that $\Pi_{0}=\text{\rm I}$ on $\ker\Box_{0}$ while $\Pi_{0}\Box_{0}=0$ so $\Pi_{0}=0$ on ${\rm Im}\,\Box_{0}$ . We obtain Part (2) from $\Omega^{\bullet}(M)=E_{0}\oplus F_{0}$ . ∎

We call $\Pi_{0}$ the base kernel projection.

4.2. The operators $\Box$ , $P$ and $L$

We define the differential operator acting on $\Omega^{\bullet}(M)$ :

\Box:=d\,\delta_{0}+\delta_{0}\,d.

Remark 4.3.

The de Rham differential $d$ will be replaced with its algebraic part $\tilde{d}$ in Section 4.4. The new construction will closely follow the steps below, as the only property that it relies on is that ${\rm gr}(d)=d_{\mathfrak{g}M}$ .

Proposition 4.4.

The operator $\Box$ commutes with $d$ and $\delta_{0}$ . It is a differential operator that preserves the degrees and the weights of forms:

\Box(\Omega^{k,\geq w}(M))\subseteq\Omega^{k,\geq w}(M).

We have

{\rm gr}(\Box)=\Box_{\mathfrak{g}M}.

Proof.

Since $d$ and $\delta_{0}$ are complexes, they commute with $\Box$ . Moreover, $d$ and $d_{0}$ increase the degrees of forms by one, while $\delta_{0}$ decreases the degrees by one. Hence, $\Box$ and $\Box_{0}$ preserve the degrees of forms. As $d$ , $d_{0}$ and $\delta_{0}$ respect the filtration, so do $\Box$ and $\Box_{0}$ . We conclude with

{\rm gr}(\Box)={\rm gr}(d){\rm gr}(\delta_{0})+{\rm gr}(\delta_{0}){\rm gr}(d)=d_{\mathfrak{g}M}d_{\mathfrak{g}M}^{t}+d_{\mathfrak{g}M}^{t}d_{\mathfrak{g}M}=\Box_{\mathfrak{g}M}.

∎

Applying Proposition 4.4 together with Proposition 2.22 to $z-\Box$ , the map

P:=\frac{1}{2\pi i}\oint_{|z|=\varepsilon}(z-\Box)^{-1}dz,

is well-defined for $|z|=\varepsilon$ small enough.

Proposition 4.5.

(1)

The map $P$ is a differential operator acting on $\Omega^{\bullet}(M)$ that respects the filtration and the degree of the forms with ${\rm gr}(P)=\Pi_{\mathfrak{g}M}$ . It commutes with $\Box$ , $d$ and $\delta_{0}$ .
(2)

We have $\Box^{N_{0}}P=0$ while $\Box^{N_{0}}$ acts on ${\rm Im}\,P$ where it is invertible. Moreover, $P$ is the projection onto

$E:=\ker\delta_{0}\cap\ker(\delta_{0}d)={\rm Im}\,P=\ker\Box^{N_{0}},$

along

$F:={\rm Im}\,\delta_{0}+{\rm Im}\,d\delta_{0}=\ker P={\rm Im}\,\Box^{N_{0}}.$

These modules satisfy ${\rm gr}(E)=E_{0}$ and ${\rm gr}(F)=F_{0}$ .

Proof.

Part (1) follows from Proposition 4.4 and applying ${\rm gr}$ . From the Cauchy residue formula, proceeding as in the proof of [FT23, Proposition 4.1], we check that $P^{2}=P$ . In other words, $P\colon\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ is a projection.

For Part (2), ${\rm gr}(\Box P)=0$ by the Cauchy formula and the properties of ${\rm gr}$ . Therefore $(\Box P)^{N_{0}}=0$ , but $(\Box P)^{N_{0}}=\Box^{N_{0}}P$ since $P$ is a projection commuting with $\Box$ . Therefore ${\rm Im}\,P\subseteq\ker\Box^{N_{0}}$ .

Naturally, $\Box$ acts on ${\rm Im}\,\Box^{N_{0}}$ which is a submodule of $\Omega^{\bullet}(M)$ that inherits a filtration. Moreover, we check

{\rm gr}({\rm Im}\,\Box^{N_{0}})={\rm Im}\,{\rm gr}(\Box^{N_{0}})={\rm Im}\,\Box_{\mathfrak{g}M}=\ker\Pi_{\mathfrak{g}M},

since $\Box_{\mathfrak{g}M}$ is symmetric and the orthogonal projection onto its kernel is $\Pi_{\mathfrak{g}M}$ , and

{\rm gr}|_{{\rm Im}\,\Box^{N_{0}}}(\Box:{\rm Im}\,\Box^{N_{0}}\to{\rm Im}\,\Box^{N_{0}})=\Box_{\mathfrak{g}M}:{\rm Im}\,\Pi_{\mathfrak{g}M}\to{\rm Im}\,\Pi_{\mathfrak{g}M}.

Adapting the proof of Proposition 2.22 to a submodule, we obtain that $\Box|_{{\rm Im}\,\Box^{N_{0}}}$ is invertible on ${\rm Im}\,\Box^{N_{0}}$ . This implies that $\ker\Box^{N_{0}}\cap{\rm Im}\,\Box^{N_{0}}=\{0\}$ and $\ker\Box^{N_{0}}\oplus{\rm Im}\,\Box^{N_{0}}=\Omega^{\bullet}(M)$ . We then readily check that $P$ is the projection onto $\ker\Box^{N_{0}}$ along ${\rm Im}\,\Box^{N_{0}}$ .

As $\Pi_{\mathfrak{g}M}d_{\mathfrak{g}M}^{t}=0$ , we have $\Pi_{0}\delta_{0}=0$ . Hence, since $P$ commutes with $\delta_{0}$ , we have

(P-\Pi_{0})\delta_{0}=P\delta_{0}=\delta_{0}P\,,

and so recursively, for any $k\geq 1$ ,

(P-\Pi_{0})^{k}\delta_{0}=P^{k}\delta_{0}=\delta_{0}P^{k}\,.

For $k\geq N_{0}$ , $(P-\Pi_{0})^{k}=0$ since ${\rm gr}(P)=\Pi_{\mathfrak{g}M}={\rm gr}(\Pi_{0})$ . As $P$ is a projection, we have obtained $0=P\delta_{0}=\delta_{0}P$ . Moreover, since $P$ commutes with $d$ , we also have $Pd\delta_{0}=dP\delta_{0}=0$ , which implies the inclusion $\ker P\supset{\rm Im}\,\delta_{0}+{\rm Im}\,d\delta_{0}=:F$ .

Since $d^{2}=0$ and $(\delta_{0})^{2}=0$ , we compute easily

\Box^{N_{0}}=(d\delta_{0})^{N_{0}}+(\delta_{0}d)^{N_{0}},

As $P$ is the projection onto $\ker\Box^{N_{0}}$ along ${\rm Im}\,\Box^{N_{0}}$ , we have

	$\displaystyle{\rm Im}\,P$	$\displaystyle=\ker\Box^{N_{0}}\supset\ker\delta_{0}\cap\ker(\delta_{0}d)=:E,$
	$\displaystyle\ker P$	$\displaystyle={\rm Im}\,\Box^{N_{0}}\subset{\rm Im}\,\delta_{0}+{\rm Im}\,(d\delta_{0})=:F\,.$

These last inclusions then imply that $\ker P=F$ , but also ${\rm Im}\,P=E$ . ∎

Note that in our construction below, we do not need the characterisations of ${\rm Im}\,P$ and $\ker P$ as $E=\ker\delta_{0}\cap\ker(\delta_{0}d)$ and $F={\rm Im}\,\delta_{0}+{\rm Im}\,d\delta_{0}$ . We will only need it when proving that this construction coincides with Rumin’s.

We follow the same strategy as in the case of homogeneous groups [FT23, Section 4.1]:

Proposition 4.6.

The differential operator $L$ acting on $\Omega^{\bullet}(M)$ and defined as

L:=P\Pi_{0}+(\text{\rm I}-P)(\text{\rm I}-\Pi_{0}),

preserves the degrees of the forms and respects the filtration with ${\rm gr}(L)=\text{\rm I}$ . It is invertible and its inverse is a differential operator acting on $\Omega^{\bullet}(M)$ . We have

PL=P\Pi_{0}=L\Pi_{0}

and this implies

P=L\Pi_{0}L^{-1}\qquad\mbox{and}\qquad(L^{-1}\Box L)\Pi_{0}=\Pi_{0}(L^{-1}\Box L).

Consequently,

F=\ker P=L(\ker\Pi_{0})L^{-1}=LF_{0}L^{-1},\qquad E={\rm Im}\,P=L({\rm Im}\,\Pi_{0})L^{-1}=LE_{0}L^{-1}.

Proof.

By Lemma 2.16, we have

{\rm gr}(L)={\rm gr}(P)\ {\rm gr}(\Pi_{0})+{\rm gr}(\text{\rm I}-P)\ {\rm gr}(\text{\rm I}-\Pi_{0})=\Pi_{\mathfrak{g}M}^{2}+(\text{\rm I}-\Pi_{\mathfrak{g}M})^{2}=\text{\rm I}.

We conclude with Proposition 2.22 and straightforward computations. ∎

4.3. The complexes $(E_{0}^{\bullet},D)$ and $(F_{0}^{\bullet},C)$

By Propositions 4.5 and 4.6, the de Rham differential $d$ commutes with $P$ , so $L^{-1}dL$ commutes with $L^{-1}PL=\Pi_{0}$ . Hence, we can decompose the differential operator $L^{-1}dL$ as

L^{-1}dL\ =\ L^{-1}dL\Pi_{0}\ +\ L^{-1}dL(\text{\rm I}-\Pi_{0})\ =\ D\ +\ C\,,

where $D$ and $C$ are the differential operators

	$\displaystyle D$	$\displaystyle:=L^{-1}dL\Pi_{0}\ =\ \Pi_{0}L^{-1}dL\Pi_{0},$
	$\displaystyle C$	$\displaystyle:=L^{-1}dL(\text{\rm I}-\Pi_{0})\ =\ (\text{\rm I}-\Pi_{0})L^{-1}dL(\text{\rm I}-\Pi_{0}).$

Since $(L^{-1}dL)^{2}=L^{-1}d^{2}L=0$ and $DC=0=CD$ , it follows that $D^{2}=C^{2}=0$ . Hence, the chain complex $L^{-1}dL$ decomposes into the direct sum of the two chain complexes: $D$ acting on $E_{0}={\rm Im}\,\Pi_{0}$ , and $C$ acting on $F_{0}=\ker\Pi_{0}$ .

Proposition 4.7.

The maps $C$ and $d_{0}$ are conjugated on $F_{0}$ :

Cg=gd_{0}\quad\mbox{on}\ F_{0}\,,

where $g:F_{0}\to F_{0}$ is the invertible operator defined as

g:=C\delta_{0}\Box_{0}^{-1}+\delta_{0}\Box_{0}^{-1}d_{0}\colon F_{0}\to F_{0}\,.

Hence, the complexes $(F_{0}^{\bullet},C)$ and $(F_{0}^{\bullet},d_{0})$ have the same cohomology.

Proof.

Note that ${\rm Im}\,d_{0}\subseteq{\rm Im}\,\Box_{0}=F_{0}$ and that $\Box_{0}$ is invertible on ${\rm Im}\,\Box_{0}=F_{0}$ , so $g$ is well-defined on $F_{0}$ .

Since ${\rm gr}(L)=\text{\rm I}$ , ${\rm gr}(d)=d_{\mathfrak{g}_{M}}$ and ${\rm gr}(\Pi_{0})=\Pi_{\mathfrak{g}M}$ , we have

{\rm gr}(C)={\rm gr}(L^{-1}dL(\text{\rm I}-\Pi_{0}))=d_{\mathfrak{g}_{M}}(\text{\rm I}-\Pi_{\mathfrak{g}M}),

and on $F_{\mathfrak{g}M}$

	$\displaystyle{\rm gr}\|_{F_{0}}(g)$	$\displaystyle={\rm gr}(C)\,d_{\mathfrak{g}_{M}}^{t}\Box_{\mathfrak{g}M}^{-1}+d_{\mathfrak{g}_{M}}^{t}\Box_{\mathfrak{g}M}^{-1}d_{\mathfrak{g}_{M}}$
		$\displaystyle=d_{\mathfrak{g}_{M}}d_{\mathfrak{g}_{M}}^{t}\Box_{\mathfrak{g}M}^{-1}+d_{\mathfrak{g}_{M}}^{t}\Box_{\mathfrak{g}M}^{-1}d_{\mathfrak{g}_{M}}$
		$\displaystyle=(d_{\mathfrak{g}_{M}}d_{\mathfrak{g}_{M}}^{t}+d_{\mathfrak{g}_{M}}^{t}d_{\mathfrak{g}_{M}})\Box_{\mathfrak{g}M}^{-1}=\text{\rm I}_{F_{\mathfrak{g}M}}\,.$

Applying the proof of Proposition 2.22 to the submodule $F_{0}$ , $g$ is invertible on ${\rm Im}\,P_{0}=F_{0}$ .

Since $C^{2}=0$ and $d_{0}^{2}=0$ , it is a straightfoward to check that $Cg=gd_{0}$ holds on $F_{0}$ . ∎

Proposition 4.8.

The differential operator $\Pi_{0}L^{-1}$ is a chain map between $(\Omega^{\bullet}(M),d)$ and $(E_{0},D)$ , that is $(\Pi_{0}L^{-1})d=D(\Pi_{0}L^{-1})$ . This chain map is homotopically invertible, with homotopic inverse given by $L$ since we have

(\Pi_{0}L^{-1})L=\Pi_{0},\quad\mbox{and}\quad\text{\rm I}-L(\Pi_{0}L^{-1})=dh+hd\,,

where $h$ is the differential operator acting on $\Omega^{\bullet}(M)$ and defined as

h:=L\,g\delta_{0}\Box_{0}^{-1}\,g^{-1}(\text{\rm I}-\Pi_{0})\,L^{-1},

with $g$ as in Proposition 4.7.

Proof.

By construction, we have $d=L(D+C)L^{-1}$ , with $D=\Pi_{0}D\Pi_{0}$ and $C=(\text{\rm I}-\Pi_{0})C(\text{\rm I}-\Pi_{0})$ , so $\Pi_{0}L^{-1}d=\Pi_{0}D\Pi_{0}L^{-1}=D\Pi_{0}L^{-1}$ .

It remains to prove the properties regarding $h$ . We first point out that $h$ makes sense because ${\rm Im}\,\delta_{0}\subset F_{0}$ and $g$ acts on $F_{0}={\rm Im}\,(\text{\rm I}-\Pi_{0})$ in an invertible way. The definitions of $h$ and $C$ together with Proposition 4.7 then yield:

	$\displaystyle L^{-1}(dh+hd)L$	$\displaystyle=L^{-1}dL\,g\delta_{0}\Box_{0}^{-1}\,g^{-1}(\text{\rm I}-\Pi_{0})+g\delta_{0}\Box_{0}^{-1}\,g^{-1}(\text{\rm I}-\Pi_{0})\,L^{-1}dL$
		$\displaystyle=C\,g\delta_{0}\Box_{0}^{-1}\,g^{-1}(\text{\rm I}-\Pi_{0})+g\delta_{0}\Box_{0}^{-1}\,g^{-1}C$
		$\displaystyle=gd_{0}\delta_{0}\Box_{0}^{-1}\,g^{-1}(\text{\rm I}-\Pi_{0})+g\delta_{0}\Box_{0}^{-1}\,d_{0}g^{-1}(\text{\rm I}-\Pi_{0})$
		$\displaystyle=g\left(d_{0}\delta_{0}\Box_{0}^{-1}+\delta_{0}\Box_{0}^{-1}d_{0}\right)g^{-1}(\text{\rm I}-\Pi_{0})=\text{\rm I}-\Pi_{0}.$

The conclusion follows. ∎

Corollary 4.9.

The cohomology of $(E_{0}^{\bullet},D)$ is linearly isomorphic to the de Rham cohomology of the manifold $M$ .

4.4. The complexes $(E_{0}^{\bullet},\tilde{D})$ and $(F_{0}^{\bullet},\tilde{C})$

In this section, we consider the complexes obtained by considering $\tilde{d}$ instead of $d$ in the construction above (see Remark 4.3). The proofs are omitted, as the arguments are essentially the same as above. We start by defining the algebraic $\tilde{\Box}$ operator acting on $\Omega^{\bullet}(M)$ :

\tilde{\Box}:=\tilde{d}\,\delta_{0}+\delta_{0}\,\tilde{d}.

It commutes with $\tilde{d}$ and $\delta_{0}$ and preserves the degrees and the weights of forms:

\tilde{\Box}(\Omega^{k,\geq w}(M))\subseteq\Omega^{k,\geq w}(M).

We have

{\rm gr}(\tilde{\Box})=\Box_{\mathfrak{g}M}.

Applying Proposition 2.22 to $z-\tilde{\Box}$ locally at $x\in M$ , together with Proposition 4.4, the map

\tilde{P}:=\frac{1}{2\pi i}\oint_{|z|=\varepsilon}(z-\tilde{\Box})^{-1}dz,

is well-defined for $|z|=\varepsilon$ small enough. $\tilde{P}$ is then an algebraic operator acting on $\Omega^{\bullet}(M)$ that respects the filtration and the degree of forms, and ${\rm gr}(\tilde{P})=\Pi_{\mathfrak{g}M}$ . It commutes with $\tilde{\Box}$ , $\tilde{d}$ and $\delta_{0}$ . We have $\tilde{\Box}^{N_{0}}\tilde{P}=0$ , while $\tilde{\Box}^{N_{0}}$ acts on ${\rm Im}\,\tilde{P}$ where it is invertible. Moreover, $\tilde{P}$ is the projection onto

\tilde{E}:=\ker\delta_{0}\cap\ker(\delta_{0}\tilde{d})={\rm Im}\,\tilde{P}=\ker\tilde{\Box}^{N_{0}},

along

\tilde{F}:={\rm Im}\,\delta_{0}+{\rm Im}\,\tilde{d}\delta_{0}=\ker\tilde{P}={\rm Im}\,\tilde{\Box}^{N_{0}}.

These modules satisfy ${\rm gr}(\tilde{E})=E_{0}$ and ${\rm gr}(\tilde{F})=F_{0}$ .

The algebraic operator $\tilde{L}$ acting on $\Omega^{\bullet}(M)$ and defined as

\tilde{L}:=\tilde{P}\Pi_{0}+(\text{\rm I}-\tilde{P})(\text{\rm I}-\Pi_{0}),

preserves the degree of forms, respects the filtration, and ${\rm gr}(\tilde{L})=\text{\rm I}$ . It is invertible and its inverse is an algebraic operator acting on $\Omega^{\bullet}(M)$ . We have

\tilde{P}\tilde{L}=\tilde{P}\Pi_{0}=\tilde{L}\Pi_{0},\qquad\tilde{P}=\tilde{L}\Pi_{0}L^{-1}\qquad\mbox{and}\qquad(\tilde{L}^{-1}\tilde{\Box}\tilde{L})\Pi_{0}=\Pi_{0}(\tilde{L}^{-1}\tilde{\Box}\tilde{L}).

Consequently,

\tilde{F}=\ker\tilde{P}=\tilde{L}(\ker\Pi_{0})\tilde{L}^{-1}=\tilde{L}F_{0}\tilde{L}^{-1},\qquad\tilde{E}={\rm Im}\,\tilde{P}=\tilde{L}({\rm Im}\,\Pi_{0})\tilde{L}^{-1}=\tilde{L}E_{0}\tilde{L}^{-1}.

We define the two differentials $\tilde{D}$ on $E_{0}={\rm Im}\,\Pi_{0}$ , and $\tilde{C}$ on $F_{0}=\ker\Pi_{0}$ via

\tilde{L}^{-1}\tilde{d}\tilde{L}=\tilde{L}^{-1}\tilde{d}\tilde{L}\Pi_{0}+\tilde{L}^{-1}\tilde{d}\tilde{L}(\text{\rm I}-\Pi_{0})=:\tilde{D}+\tilde{C}

The maps $\tilde{C}$ and $d_{0}$ are conjugated on $F_{0}$ :

\tilde{C}\tilde{g}=\tilde{g}d_{0}\quad\mbox{on}\ F_{0}\,,

where $\tilde{g}:F_{0}\to F_{0}$ is the invertible operator defined as

\tilde{g}:=\tilde{C}\delta_{0}\Box_{0}^{-1}+\delta_{0}\Box_{0}^{-1}d_{0}\colon F_{0}\to F_{0}\,.

The complexes $(F_{0}^{\bullet},\tilde{C})$ and $(F_{0}^{\bullet},d_{0})$ have the same cohomology.

The differential operator $\Pi_{0}\tilde{L}^{-1}$ is a chain map between $(\Omega^{\bullet}(M),\tilde{d})$ and $(E_{0},\tilde{D})$ , that is $(\Pi_{0}\tilde{L}^{-1})\tilde{d}=\tilde{D}(\Pi_{0}\tilde{L}^{-1})$ . This chain map is homotopically invertible, with homotopic inverse given by $\tilde{L}$ since we have

(\Pi_{0}\tilde{L}^{-1})\tilde{L}=\Pi_{0},\quad\mbox{and}\quad\text{\rm I}-\tilde{L}(\Pi_{0}\tilde{L}^{-1})=\tilde{d}\tilde{h}+\tilde{h}\tilde{d}\,,

where $\tilde{h}$ is the algebraic operator acting on $\Omega^{\bullet}(M)$ and defined as

\tilde{h}:=\tilde{L}\,\tilde{g}\delta_{0}\Box_{0}^{-1}\,\tilde{g}^{-1}(\text{\rm I}-\Pi_{0})\,\tilde{L}^{-1}.

Consequently, the cohomology of $(E_{0}^{\bullet},\tilde{D})$ is linearly isomorphic to the cohomology of $\tilde{d}$ of the manifold $M$ .

4.5. Equivalent constructions for the subcomplexes $(E_{0}^{\bullet},D)$ and $(E_{0}^{\bullet},\tilde{D})$

Here, we present an interpretation of Rumin’s construction [Rum99, Rum05] of the complex that bears his name. In our presentation, we highlight the difference between objects living on the manifold and their osculating counterparts.

We first define the map

(4.1)

d_{0}^{-1}:=\Phi^{-1}\circ d_{\mathfrak{g}M}^{-1}\circ\Phi\colon\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)\,,

which is defined via the partial inverse $d_{\mathfrak{g}M}^{-1}$ of the osculating differential $d_{\mathfrak{g}M}$ (see Section 3.1.4). From the properties of $d_{\mathfrak{g}M}^{-1}$ , we see that $(d_{0}^{-1})^{2}=0$ , it respects the filtration and decreases the degree by one:

\forall\,k,w\in\mathbb{N}_{0}\qquad d_{0}^{-1}(\Omega^{k+1,\geq w}(M))\subseteq\Omega^{k,\geq w}(M)\,.

We have

\ker d_{0}^{t}=\ker d_{0}^{-1}\quad\mbox{and}\quad{\rm Im}\,d_{0}^{t}={\rm Im}\,d_{0}^{-1},

E_{0}=\ker d_{0}\cap\ker d_{0}^{-1}\quad\mbox{and}\quad F_{0}={\rm Im}\,d_{0}+{\rm Im}\,d_{0}^{-1}.

Moreover,

(4.2)

\Pi_{0}=\text{\rm I}-d_{0}^{-1}d_{0}-d_{0}d_{0}^{-1},

and we have

d_{0}^{-1}\Pi_{0}=\Pi_{0}d_{0}^{-1}=0\ ,\quad d_{0}\Pi_{0}=\Pi_{0}d_{0}=0\,.

Lemma 4.10.

Let us consider the differential operators acting on $\Omega^{\bullet}(M)$ defined by:

b:=d_{0}^{-1}d_{0}-d_{0}^{-1}d=-d_{0}^{-1}(d-d_{0})\quad\mbox{and}\quad b_{1}:=d_{0}d_{0}^{-1}-dd_{0}^{-1}=-(d-d_{0})d_{0}^{-1}.

(1)

The maps $b$ and $b_{1}$ are nilpotent. Consequently, $\text{\rm I}-b$ and $\text{\rm I}-b_{1}$ are invertible, and $(\text{\rm I}-b)^{-1}$ and $(\text{\rm I}-b_{1})^{-1}$ are well-defined differential operators acting on $\Omega^{\bullet}(M)$ . We have:

$bd_{0}^{-1}=d_{0}^{-1}b_{1},\quad\mbox{and}\quad(\text{\rm I}-b)^{-1}d_{0}^{-1}=d_{0}^{-1}(\text{\rm I}-b_{1})^{-1}.$

(2)

The differential operator $\Pi\colon\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ defined as

\displaystyle\Pi

\displaystyle:=(\text{\rm I}-b)^{-1}d_{0}^{-1}d+d(\text{\rm I}-b)^{-1}d_{0}^{-1}=d_{0}^{-1}(\text{\rm I}-b_{1})^{-1}d+dd_{0}^{-1}(\text{\rm I}-b_{1})^{-1}\,,

is the projection onto

F:={\rm Im}\,d_{0}^{-1}+{\rm Im}\,dd_{0}^{-1}={\rm Im}\,d_{0}^{t}+{\rm Im}\,dd_{0}^{t}\,,

along

E:=\ker d_{0}^{-1}\cap\ker(d_{0}^{-1}d)=\ker d_{0}^{t}\cap\ker(d_{0}^{t}d)\,.

Proof.

By Proposition 3.3 and Lemma 2.19, $d$ and $d_{0}$ increase the degree of forms by one, respect the filtration, and ${\rm gr}(d)=d_{\mathfrak{g}M}={\rm gr}(d_{0})$ . As $d_{0}^{-1}$ decreases the degree of forms by one, $b$ respects the filtration, and ${\rm gr}(b)=-{\rm gr}(d_{0}^{-1})\ {\rm gr}(d-d_{0})=0$ . Consequently, it is nilpotent with $b^{N_{0}}=0$ . The proof then follows as in [FT23, Lemma 3.9]. ∎

Corollary 4.11.

The projections $\Pi$ and $P$ constructed in Lemma 4.10 and Proposition 4.5 respectively coincide.

Following Rumin’s notation, the two projections onto $F$ and $E$ along $E$ and $F$ are denoted respectively as

\Pi_{F}:=\Pi\qquad\mbox{and}\qquad\Pi_{E}:=\text{\rm I}-\Pi\,.

We then obtain Rumin’s construction and its equivalence with the one presented in this paper.

Theorem 4.12.

(1)

(M. Rumin) The de Rham complex $(\Omega^{\bullet}(M),d)$ splits into two subcomplexes $(E^{\bullet},d)$ and $(F^{\bullet},d)$ . Moreover, the differential operator defined as

$d_{c}:=\Pi_{0}d\,\Pi_{E}\,\Pi_{0}\colon\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)\ ,$

satisfies

$d_{c}^{2}=0\quad,\quad d_{c}(\Omega^{k}(M))\subset\Omega^{k+1}(M)\,.$

Moreover, we have

$\Pi_{E}=\Pi_{E}\Pi_{0}\Pi_{E}\quad\mbox{and}\quad\Pi_{0}\Pi_{E}\Pi_{0}=\Pi_{0}.$

and so $E_{0}={\rm Im}\,\Pi_{0}=\Pi_{0}E$ , and the complex $(E^{\bullet},d)$ is conjugated to $(E_{0}^{\bullet},d_{c})$ via $\Pi_{0}$ .
(2)

The maps $d_{c}$ and $D$ coincide, i.e. $d_{c}=D$ .

Proof.

Adapting the proof of [FT23, Lemma 3.10], it is easy to check that the operators $d_{0}^{-1}$ , $\Pi_{E}$ and $\Pi_{F}$ satisfy the following properties:

1.

$d_{0}^{-1}\Pi_{E}=\Pi_{E}d_{0}^{-1}=0$ ;
2.

$d\Pi_{F}=\Pi_{F}d$ , and $d\Pi_{E}=\Pi_{E}d$ ;
3.

$\Pi_{E}(\text{\rm I}-\Pi_{0})\Pi_{E}=0$ ;
4.

on $\Omega^{k}(M)$ , we have $\Pi_{F}^{t}=(-1)^{k(n-k)}\star\Pi_{F}\star$ and $\Pi_{E}^{t}=(-1)^{k(n-k)}\star\Pi_{E}\star$ .

These readily imply Part (1) and Part (2), after adapting the proofs of Theorems 3.11 and 4.9 in [FT23] respectively. ∎

This equivalent construction also works if we replace $d$ with $\tilde{d}$ , yielding a complex $(E_{0}^{\bullet},\tilde{d}_{c})$ which coincides with $\tilde{D}$ .

4.6. Case of regular subRiemannian manifolds

Regular subRiemannian manifolds are filtered manifolds (see below). If we equip such a manifold with a Riemannian metric, then our construction applies and we obtain two complexes $(E_{0}^{\bullet},D)$ and $(E_{0}^{\bullet},\tilde{D})$ . Here, we show that if the Riemannian metric is compatible with the subRiemannian structure as explained in Section 4.6.2 below, then $(E_{0}^{\bullet},D)$ coincides at least locally with what is now customarily referred to as the Rumin complex. We would like to stress that, when introducing what Rumin calls the “Carnot complex of an $E_{0}$ -regular $CC$ -structure” in [Rum05], he only briefly mentions the potential impact that the choice of a Riemannian metric can have on the resulting subcomplex of $M$ because he considers only local or osculating objects. In particular, he does not address any compatibility conditions between the Riemannian and subRiemannian structures, although it seems to be an important ingredient in the construction. We hope that our approach will lead to a better understanding of the constructed subcomplexes.

4.6.1. The osculating metric of a regular subRiemannian manifold

Recall that a subRiemannian manifold is a smooth manifold $M$ equipped with a bracket generating distribution $\mathcal{D}\subset TM$ and with a metric $g_{\mathcal{D}}$ on $\mathcal{D}$ . Let $\Gamma^{1}=\Gamma(\mathcal{D})$ be the set of smooth sections of $\mathcal{D}$ , and $\Gamma^{i}:=[\Gamma^{1},\Gamma^{i-1}]+\Gamma^{i-1}$ , for $i>1.$ As $\mathcal{D}$ is bracket generating, there exists $i$ such that $\Gamma^{i}=TM$ , and we denote by $s$ the smallest such integer $i$ . If, for every $i=1,\ldots,s$ , there exists a subbundle $H^{i}\subset TM$ for which $\Gamma^{i}=\Gamma(H^{i})$ is the set of smooth sections on $H^{i}$ , then $M$ is said to have an regular subRiemannian structure. Clearly, the $H^{i}$ s provide the structure of filtered manifold.

Consider a regular subRiemannian manifold $M$ as above. In this case, the osculating Lie algebras are stratified

\displaystyle\mathfrak{g}_{x}M=\oplus_{i=1}^{s}\mathfrak{g}_{i,x},\qquad\mathfrak{g}_{i,x}=H^{i}_{x}/H^{i+1}_{x}\ ,\ \forall\,x\in M\,,

and the subspaces $\mathfrak{g}_{i,x}\subset\mathfrak{g}_{x}M$ are given by imposing $\mathfrak{g}_{1}:=\mathcal{D}$ , and

\mathfrak{g}_{2,x}=[\mathfrak{g}_{1,x},\mathfrak{g}_{1,x}]_{\mathfrak{g}_{x}M},\quad\mathfrak{g}_{3,x}=[\mathfrak{g}_{1,x},\mathfrak{g}_{2,x}]_{\mathfrak{g}_{x}M},\ldots,\ \mathfrak{g}_{i,x}=[\mathfrak{g}_{1,x},[\mathfrak{g}_{i-1,x}]_{\mathfrak{g}_{x}M},\ldots

The metric $g_{\mathcal{D}}$ on $\mathcal{D}$ naturally induces a scalar product on each osculating Lie algebra fibre $\mathfrak{g}_{x}M$ [Mon02, p. 188]. Let us briefly recall its construction.

At every point $x\in M$ , the map

(\mathfrak{g}_{1,x})^{j}=\mathfrak{g}_{1,x}\times\ldots\times\mathfrak{g}_{1,x}\to\mathfrak{g}_{j,x}\ ,\ \,(V_{1},\ldots,V_{j})\longmapsto[V_{1},[V_{2},[\cdots,V_{j}]_{\mathfrak{g}_{x}M}]_{\mathfrak{g}_{x}M}\cdots]_{\mathfrak{g}_{x}M}\ ,

is $j$ -linear and surjective. Hence, the metric $g_{\mathcal{D}}$ on $\mathfrak{g}_{1}=H^{1}=\mathcal{D}$ induces a scalar product on $(\mathfrak{g}_{1,x})^{j}$ and then on $\mathfrak{g}_{j,x}$ (which is the image of the above map, and therefore inherits the scalar product from the orthogonal complement of the kernel). Constructing this for $j=2,\ldots,s$ , we obtain a scalar product $g_{\mathfrak{g}_{x}M}$ on $\mathfrak{g}_{x}M$ . By construction, the dependence in $x$ is smooth, and $(g_{\mathfrak{g}M})(x):=g_{\mathfrak{g}_{x}M}$ , $x\in M$ , defines an element $g_{\mathfrak{g}M}\in\Gamma({\rm Sym}(\mathfrak{g}M\otimes\mathfrak{g}M))$ . Therefore, $g_{\mathfrak{g}M}$ is a metric on the osculating bundle $\mathfrak{g}M$ of Lie algebras. We call $g_{\mathfrak{g}M}$ the osculating metric induced by $g_{\mathcal{D}}$ .

4.6.2. Compatible Riemannian metrics

We now assume that, in addition to being a regular subRiemannian manifold, $M$ is also equipped with a Riemannian metric, that is, with a metric $g_{TM}$ on its tangent bundle $TM$ . As already seen in Sections 2.1.10 and 2.5.1, $g_{TM}$ will also induce a metric $g_{{\rm gr}(TM)}$ on ${\rm gr}(TM)\cong\mathfrak{g}M$ . The two metrics $g_{\mathfrak{g}M}$ and $g_{{\rm gr}(TM)}$ will be different in general.

Definition 4.13.

A Riemannian metric $g_{TM}$ is compatible with the subRiemannian structure of a regular subRiemannian manifold $M$ (or compatible for short) when the two metrics $g_{{\rm gr}(TM)}$ and $g_{\mathfrak{g}M}$ on the osculating Lie algebra bundle $\mathfrak{g}M\cong{\rm gr}(TM)$ coincide.

By construction, $g_{\mathfrak{g}M}$ coincides on $\mathcal{D}=H^{1}=\mathfrak{g}_{1}$ with $g_{\mathcal{D}}$ . Hence, a necessary condition for $g_{TM}$ to be compatible is for $g_{TM}$ to coincide with $g_{\mathcal{D}}$ on $\mathcal{D}$ . However, it is not sufficient generally. For example, let us consider the 6-dimensional free nilpotent Lie group of rank 3 and step 2, denoted by $N_{6,3,6}$ in [LDT22]. Keeping the notation of [LDT22], if we take $\Theta=\{\theta^{1},\theta^{2},\theta^{3},\theta^{4},\theta^{5},\theta^{6}\}$ and $\hat{\Theta}=\{\theta^{1},\theta^{2},\theta^{3},\theta^{4},\theta^{4}+\theta^{5},\theta^{5}+\theta^{6}\}$ as global left-invariant orthogonal coframes of $TN_{6,3,6}$ , then the corresponding subcomplexes $(E_{0}^{\bullet},D)$ and $(\hat{E}_{0}^{\bullet},\hat{D})$ do not coincide [FT24]. For clarity, we stress that $E_{0}^{\bullet}$ and $\hat{E}_{0}^{\bullet}$ are isomorphic subspaces of $\Omega^{\bullet}(N_{6,3,6})$ , and this is necessarily true in general once we assume $M$ is a regular subRiemannian manifold. However, they need not be the same subspace, and in this explicit example they are not (it is sufficient to compare $E_{0}^{2}$ and $\hat{E}_{0}^{2}$ ). In other words, just extending the metric $g_{\mathcal{D}}$ from $\mathcal{D}$ to an arbitrary Riemannian metric on $TM$ is not sufficient to ensure compatibility. However, compatible metrics can always be constructed.

Lemma 4.14.

Let $M$ be a (second countable) regular subRiemannian manifold. We can always construct a compatible Riemannian metric.

Proof.

Consider a sequence of non-negative functions $\varphi_{i}\in C_{c}^{\infty}(M)$ , $i\in\mathbb{N}$ , such that the sum $\sum_{i\in\mathbb{N}}\varphi_{i}$ on $M$ is locally finite and constantly equal to 1. We may assume that the support of each $\varphi_{i}$ is small enough so that it is included in an open subset $U_{i}\subset M$ where a frame $\mathbb{X}_{i}$ of $TU_{i}$ exists. After a Graham-Schmidt procedure, we may assume that each $\mathbb{X}_{i}$ is such that the corresponding frame $\langle\mathbb{X}_{i}\rangle$ of the vector bundle $\mathfrak{g}M|_{U_{i}}$ is $g_{\mathfrak{g}M}$ -orthonormal. Denoting by $g_{i}$ the corresponding metric on $U_{i}$ such that $\mathbb{X}_{i}$ is orthonormal, one can easily check that $g=\sum_{i\in\mathbb{N}}\varphi_{i}g_{i}$ is a compatible Riemannian metric. ∎

4.6.3. The base differential and codifferential associated with a compatible Riemannian metric

Let $M$ be a regular subRiemannian manifold equipped with a Riemannian metric $g_{TM}$ . The compatibility implies that the scalar product on $\wedge^{\bullet}{\rm gr}(T_{x}^{*}M)$ (defined as in Section 2.5.1 using $g_{TM}$ ) coincides with the scalar product on $\wedge^{\bullet}\mathfrak{g}_{x}^{*}M$ induced by the osculating metric $g_{\mathfrak{g}M}$ . Hence, the map $\Phi$ identifying forms and osculating forms and the objects built upon it (for instance the base differential and co-differential $d_{0}$ and $\delta_{0}$ ) are defined globally as objects leaving on the manifolds. These removes ambiguities and improves on earlier constructions [Rum90, Rum99, Rum05].

Appendix A Case of a three-dimensional contact manifold (locally)

On a three-dimensional contact manifold, the existence of Darboux coordinates implies that the manifold is filtered and can be locally described as follows. The manifold $M$ is equipped with a frame $X,Y,T$ satisfying

	$\displaystyle[X,Y]$	$\displaystyle=c_{0}T+c_{1}X+c_{2}Y,$
	$\displaystyle[X,T]$	$\displaystyle=c_{3}X+c_{4}Y,$
	$\displaystyle[Y,T]$	$\displaystyle=c_{5}X+c_{6}Y,$

with $c_{j}\in C^{\infty}(M)$ , $j=0,\ldots,6$ , with $c_{0}$ nowhere vanishing.

The associated free basis $\Theta^{\bullet}$ of $\Omega^{\bullet}(M)$ is given by

	$\displaystyle k=0$	$\displaystyle\qquad 1(\geq 0),$
	$\displaystyle k=1$	$\displaystyle\qquad X^{}(\geq 1),\ Y^{}(\geq 1),\ T^{*}(\geq 2),$
	$\displaystyle k=2$	$\displaystyle\qquad X^{}\wedge Y^{}(\geq 2),\ X^{}\wedge T^{}(\geq 3),\ Y^{}\wedge T^{}(\geq 3),$
	$\displaystyle k=3$	$\displaystyle\qquad{\rm vol}=X^{}\wedge Y^{}\wedge T^{*}(\geq 4).$

The numbers in parenthesis refer to the $\geq$ weight of the form.

Description of $\tilde{d}$

For $k=0,3$ , $\tilde{d}^{(k)}=0$ . For $k=1,2$ , let us describe $\tilde{d}$ in matrix form with respect to the canonical basis described above.

{\rm Mat}(\tilde{d}^{(1)})=\left(\begin{array}[]{ccc}-c_{1}&-c_{2}&-c_{0}\\ -c_{3}&-c_{4}&0\\ -c_{5}&-c_{6}&0\end{array}\right),\qquad{\rm Mat}(\tilde{d}^{(2)})=\left(\begin{array}[]{ccc}c_{3}+c_{6}&-c_{1}&-c_{2}\end{array}\right).

Description of $d$

For $k=3$ , we have $d^{(3)}=0$ , while for $k=0$ , we have $df(V)=Vf$ , $f\in C^{\infty}(M)$ . Hence,

	$\displaystyle{\rm Mat}(d^{(0)})$	$\displaystyle=\left(\begin{array}[]{c}X\\ Y\\ T\end{array}\right)$
	$\displaystyle{\rm Mat}(d^{(1)})$	$\displaystyle=\left(\begin{array}[]{ccc}-Y&X&0\\ -T&0&X\\ 0&-T&Y\end{array}\right)+{\rm Mat}(\tilde{d}^{(1)})$
	$\displaystyle{\rm Mat}(d^{(2)})$	$\displaystyle=\left(\begin{array}[]{ccc}T&-Y&X\end{array}\right)+{\rm Mat}(\tilde{d}^{(2)})$

Description of $d_{\mathfrak{g}M}$

The osculating group is isomorphic to the Heisenberg group at any point $p\in M$ :

[\langle X\rangle_{p},\langle Y\rangle_{p}]_{\mathfrak{g}_{x}M}=c_{0}(p)\langle T\rangle_{p},\qquad[\langle X\rangle_{p},\langle T\rangle_{p}]_{\mathfrak{g}_{x}M}=0=[\langle Y\rangle_{p},\langle T\rangle_{p}]_{\mathfrak{g}_{x}M}.

The associated free basis of $G\Omega^{\bullet}(M)$ is

	$\displaystyle k=0$	$\displaystyle\qquad 1(=0),$
	$\displaystyle k=1$	$\displaystyle\qquad\langle X^{}\rangle(=1),\ \langle Y^{}\rangle(=1),\ \langle T^{*}\rangle(=2),$
	$\displaystyle k=2$	$\displaystyle\qquad\langle X^{}\rangle\wedge\langle Y^{}\rangle(=2),\ \langle X^{}\rangle\wedge\langle T^{}\rangle(=3),\ \langle Y^{}\rangle\wedge\langle T^{}\rangle(=3),$
	$\displaystyle k=3$	$\displaystyle\qquad\langle X^{}\rangle\wedge\langle Y^{}\rangle\wedge\langle T^{*}\rangle(=4).$

The number in parenthesis refers to the weight of the osculating form.

For $k=0,2,3$ , $d_{\mathfrak{g}M}^{(k)}=0$ . For $k=1$ , let us describe $d_{\mathfrak{g}M}$ in matrix form with respect to the canonical basis described above.

{\rm Mat}(d_{\mathfrak{g}M}^{(1)})=\left(\begin{array}[]{ccc}0&0&-c_{0}\\ 0&0&0\\ 0&0&0\end{array}\right).

Description of $d_{0}$ and $d_{0}^{t}$

For $k=0,2,3$ , $d_{0}^{(k)}=0$ while for $k=1$ , the description of $d_{0}^{(1)}$ in matrix form with respect to the canonical basis of $\Omega^{\bullet}(M)$ given above coincides with ${\rm Mat}(d_{\mathfrak{g}M}^{(1)})$ . For $k=0,1,3$ , $d_{0}^{(k,t)}=0$ . For $k=2$ , the matrix description of $d_{0}^{t}$ coincides with the transpose of ${\rm Mat}(d_{\mathfrak{g}M}^{(1)})$ . Consequently,

{\rm Mat}(d_{0}^{(1)})=\left(\begin{array}[]{ccc}0&0&-c_{0}\\ 0&0&0\\ 0&0&0\end{array}\right)\ ,\quad{\rm Mat}(d_{0}^{(1,t)})=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&0\\ -c_{0}&0&0\end{array}\right).

Description of $\Box_{0}$ , $P_{0}$

For $k=0,3$ , $\Box^{(k)}_{0}=0$ , while $\Box^{(1)}_{0}=d_{0}^{(1,t)}d_{0}^{(1)}$ and $\Box^{(2)}_{0}=d_{0}^{(1)}d_{0}^{(1,t)}$ are represented by the following diagonal matrices:

	$\displaystyle{\rm Mat}\big{(}\Box_{0}^{(1)}\big{)}$	$\displaystyle={\rm Mat}\big{(}d_{0}^{(1,t)}\big{)}{\rm Mat}\big{(}d_{0}^{(1)}\big{)}={\rm diag}\big{(}0,0,c_{0}^{2}\big{)}\ ,$
	$\displaystyle{\rm Mat}\big{(}\Box_{0}^{(2)}\big{)}$	$\displaystyle={\rm Mat}\big{(}d_{0}^{(1)}\big{)}{\rm Mat}\big{(}d_{0}^{(1,t)}\big{)}={\rm diag}\big{(}c_{0}^{2},0,0\big{)}\ .$

Consequently, $\Pi_{0}^{(k)}$ is the identity for $k=0,3$ , while for $k=1,2$ ,

{\rm Mat}\big{(}\Pi_{0}^{(1)}\big{)}={\rm diag}(1,1,0)\qquad\mbox{and}\qquad{\rm Mat}\big{(}\Pi_{0}^{(2)}\big{)}={\rm diag}(0,1,1)\ .

Consequently, $E_{0}^{(0)}=\Omega^{0}(M)=\Phi^{-1}\big{(}\langle 1\rangle\big{)}$ and $E_{0}^{(3)}=\Omega^{3}(M)=\Phi^{-1}\big{(}\langle X^{*}\wedge Y^{*}\wedge T^{*}\rangle\big{)}$ while

E_{0}^{(1)}=\Phi^{-1}\big{(}\langle X^{*}\rangle\oplus\langle Y^{*}\rangle\big{)}\quad\mbox{and}\quad E_{0}^{(2)}=\Phi^{-1}\big{(}\langle X^{*}\wedge T^{*}\rangle\oplus\langle Y^{*}\wedge T^{*}\rangle\big{)}.

Description of $\Box$ , $P$ , $L$

For $k=0,3$ , we have $\Box^{(k)}=0$ , while for $k=1,2$

	$\displaystyle{\rm Mat}\big{(}\Box^{(1)}\big{)}$	$\displaystyle={\rm Mat}\big{(}d_{0}^{(1,t)}\big{)}{\rm Mat}\big{(}d^{(1)}\big{)}=c_{0}\left(\begin{array}[]{ccc}0&0&0\\ 0&0&0\\ Y+c_{1}&-X+c_{2}&c_{0}\end{array}\right),$
	$\displaystyle{\rm Mat}\big{(}\Box^{(2)}\big{)}$	$\displaystyle={\rm Mat}\big{(}d^{(1)}\big{)}{\rm Mat}\big{(}d_{0}^{(1,t)}\big{)}=-c_{0}\left(\begin{array}[]{ccc}-c_{0}&0&0\\ X&0&0\\ Y&0&0\end{array}\right).$

Consequently, $P^{(k)}$ is the identity on $\Omega^{k}(M)$ for $k=0,3$ . Since $\big{(}(z-\Box_{0})(\Box_{0}-\Box)\big{)}^{j}=0$ for $j>1$ , the formulae for Cauchy residues and the von Neumann series simplify into

P=\Pi_{0}+\Pi_{0}(\Box_{0}-\Box)c_{0}^{-2}{\rm pr}_{c_{0}^{2}}+c_{0}^{-2}{\rm pr}_{c_{0}^{2}}(\Box_{0}-\Box)\Pi_{0}\ ,

where ${\rm pr}_{c_{0}^{2}}$ denotes the projection onto the $c_{0}^{2}$ -eigenspace of $\Box_{0}$ . Hence, for $k=1,2$ ,

	$\displaystyle{\rm Mat}\big{(}P^{(1)}\big{)}$	$\displaystyle={\rm diag}(1,1,0)-c_{0}^{-1}\left(\begin{array}[]{ccc}0&0&0\\ 0&0&0\\ Y+c_{1}&-X+c_{2}&0\end{array}\right)\ ,$
	$\displaystyle{\rm Mat}\big{(}P^{(2)}\big{)}$	$\displaystyle={\rm diag}(0,1,1)+c_{0}^{-1}\left(\begin{array}[]{ccc}0&0&0\\ X&0&0\\ Y&0&0\end{array}\right)\ .$

We now describe

L=P\Pi_{0}+(\text{\rm I}-P)(\text{\rm I}-\Pi_{0})=\text{\rm I}+(P-\Pi_{0})(-\text{\rm I}+2\Pi_{0}).

For $k=0,3$ , $L^{(k)}$ is the identity on $\Omega^{k}(M)$ while for $k=1,2$ ,

	$\displaystyle{\rm Mat}\big{(}L^{(1)}\big{)}$	$\displaystyle=\text{\rm I}_{3}-c_{0}^{-1}\left(\begin{array}[]{ccc}0&0&0\\ 0&0&0\\ Y+c_{1}&-X+c_{2}&0\end{array}\right)$
	$\displaystyle{\rm Mat}\big{(}L^{(2)}\big{)}$	$\displaystyle=\text{\rm I}_{3}-c_{0}^{-1}\left(\begin{array}[]{ccc}0&0&0\\ X&0&0\\ Y&0&0\end{array}\right),$

where $\text{\rm I}_{3}={\rm diag}(1,1,1)$ denotes the 3-by-3 identity matrix.

Computation of $D$

We now describe

D^{(k)}=\big{(}L^{(k+1)}\big{)}^{-1}d^{(k)}L^{(k)}\Pi_{0}^{(k)},

for $k=0,1,2$ .

	$\displaystyle{\rm Mat}\big{(}D^{(0)}\big{)}$	$\displaystyle=\left(\begin{array}[]{c}X\\ Y\\ T+c_{0}^{-1}((Y+c_{1})X+(-X+c_{2})Y)\end{array}\right)=\left(\begin{array}[]{c}X\\ Y\\ 0\end{array}\right),$
	$\displaystyle{\rm Mat}(D^{(1)})$	$\displaystyle=\left(\begin{array}[]{ccc}0&0&0\\ -T-c_{3}-c_{0}^{-1}X(Y+c_{1})&-c_{4}+c_{0}^{-1}X(X-c_{2})&0\\ -c_{5}-c_{0}^{-1}Y(Y+c_{1})&-T-c_{6}+Yc_{0}^{-1}(X-c_{2})&0\end{array}\right),$
	$\displaystyle{\rm Mat}(D^{(2)})$	$\displaystyle=\left(\begin{array}[]{ccc}0&-Y-c_{1}&X-c_{2}\end{array}\right).$

The zeros in the matrices above illustrate $D$ acting on $E_{0}$ while being trivial on $F_{0}$ .

Description of $\tilde{\Box}$ , $\tilde{P}$ , $\tilde{L}$

For $k=0,3$ , we have $\tilde{\Box}^{(k)}=0$ , while for $k=1,2$

	$\displaystyle{\rm Mat}\left(\tilde{\Box}^{(1)}\right)=$	$\displaystyle{\rm Mat}\big{(}d_{0}^{(1,t)}\big{)}{\rm Mat}\big{(}\tilde{d}^{(1)}\big{)}=c_{0}\left(\begin{array}[]{ccc}0&0&0\\ 0&0&0\\ c_{1}&c_{2}&c_{0}\end{array}\right)\ ,$
	$\displaystyle{\rm Mat}\big{(}\tilde{\Box}^{(2)}\big{)}=$	$\displaystyle{\rm Mat}\big{(}\tilde{d}^{(1)}\big{)}{\rm Mat}\big{(}d_{0}^{(1,t)}\big{)}={\rm diag}\big{(}c_{0}^{2},0,0\big{)}\ .$

Consequently, $\tilde{P}^{(k)}$ is the identity on $\Omega^{k}(M)$ for $k=0,3$ . Since $\big{(}(z-\Box_{0})(\Box_{0}-\tilde{\Box})\big{)}^{j}=0$ for $j>1$ , the formulae for Cauchy residues and the von Neumann series simplify into

\displaystyle\tilde{P}=\Pi_{0}+\Pi_{0}(\Box_{0}-\tilde{\Box})c_{0}^{-2}{\rm pr}_{c_{0}^{2}}+c_{0}^{-2}{\rm pr}_{c_{0}^{2}}(\Box_{0}-\tilde{\Box})\Pi_{0}\ ,

where ${\rm pr}_{c_{0}^{2}}$ denotes the projection onto the $c_{0}^{2}$ -eigenspace of $\Box_{0}$ . Hence, for $k=1,2$ ,

	$\displaystyle{\rm Mat}\big{(}\tilde{P}^{(1)}\big{)}=$	$\displaystyle{\rm diag}(1,1,0)-c_{0}^{-1}\left(\begin{array}[]{ccc}0&0&0\\ 0&0&0\\ c_{1}&c_{2}&0\end{array}\right)$
	$\displaystyle{\rm Mat}\big{(}\tilde{P}^{(2)}\big{)}=$	$\displaystyle{\rm diag}(0,1,1)\ .$

We now describe

\displaystyle\tilde{L}=\tilde{P}\Pi_{0}+(\text{\rm I}-\tilde{P})(\text{\rm I}-\Pi_{0})=\text{\rm I}+(\tilde{P}-\Pi_{0})(-\text{\rm I}+2\Pi_{0})\ .

For $k=0,3$ , $\tilde{L}^{(k)}$ is the identity on $\Omega^{k}(M)$ while for $k=1,2$ ,

\displaystyle{\rm Mat}\big{(}\tilde{L}^{(1)}\big{)}=\text{\rm I}_{3}-c_{0}^{-1}\left(\begin{array}[]{ccc}0&0&0\\ 0&0&0\\ c_{1}&c_{2}&0\end{array}\right)\ ,\ {\rm Mat}\big{(}\tilde{L}^{(2)}\big{)}=\text{\rm I}_{3}\ ,

where $\text{\rm I}_{3}={\rm diag}(1,1,1)$ denotes the 3-by-3 identity matrix.

Computation of $\tilde{D}$

We now describe

\displaystyle\tilde{D}^{(k)}=\big{(}L^{(k+1)}\big{)}^{-1}\tilde{d}^{(k)}\tilde{L}^{(k)}\Pi_{0}^{(k)}

for $k=0,1,2$ .

{\rm Mat}\big{(}\tilde{D}^{(0)}\big{)}=\left(\begin{array}[]{c}0\\ 0\\ 0\end{array}\right),\quad{\rm Mat}\big{(}\tilde{D}^{(1)}\big{)}=\left(\begin{array}[]{ccc}0&0&0\\ -c_{3}&-c_{4}&0\\ -c_{5}&-c_{6}&0\end{array}\right),

{\rm Mat}\big{(}\tilde{D}^{(2)}\big{)}=\left(\begin{array}[]{ccc}0&-c_{1}&-c_{2}\end{array}\right).

Appendix B Examples

In this appendix, we give two examples of a differential operator $D\colon\Omega^{\bullet}(M)\to\Omega^{\bullet}(M)$ that respects the filtration (2.11) whose transpose $D^{t}$ does not necessarily respect the filtration.

Example B.1.

We consider the 3-dimensional Heisenberg group $\mathbb{H}$ , and the canonical basis $X,Y,T$ of its Lie algebra $\mathfrak{h}$ . The only non-trivial bracket is $[X,Y]=T$ . Identifying $\mathfrak{h}$ with the space of left-invariant vector fields on $\mathbb{H}$ , we obtain a filtration of the tangent bundle of $\mathbb{H}$ in subbundles:

\{0\}=H^{0}\subset H^{1}:={\rm span}_{\mathbb{R}}\{X,Y\}\subset H^{2}:={\rm span}_{\mathbb{R}}\{X,Y,T\}=T\mathbb{H}^{1}\,.

The dual of the left-invariant frame $\mathbb{X}:=(X,Y,T)$ of $T\mathbb{H}$ yields the global coframe $\Theta:=(X^{\ast},Y^{\ast},T^{\ast})$ (as well as the natural linear isomorphism between $\Omega^{\bullet}(\mathbb{H})$ and $C^{\infty}(\mathbb{H})\otimes\wedge^{\bullet}\mathfrak{h}^{*}$ ).

Let us now consider the de Rham differential $d\colon\Omega^{\bullet}(\mathbb{H})\to\Omega^{\bullet}(\mathbb{H})$ acting on the 1-form $f\,T^{\ast}\in\Omega^{1,\geq 2}(\mathbb{H})$ with $f\in C^{\infty}(\mathbb{H})$ :

\displaystyle d(fT^{\ast})=df\wedge T^{\ast}+f\,dT^{\ast}=Xf\,(X^{\ast}\wedge T^{\ast})+Yf\,(Y^{\ast}\wedge T^{\ast})-f\,(X^{\ast}\wedge Y^{\ast})\in\Omega^{2,\geq 2}(\mathbb{H})\,.

If we now consider its formal transpose $d^{t}=(-1)^{k\cdot 3+1}\star d\star\colon\Omega^{k+1}(\mathbb{H})\to\Omega^{k}(\mathbb{H})$ acting on the 2-form $g\,(X^{\ast}\wedge Y^{\ast})\ \in\ \Omega^{2,\geq 2}(\mathbb{H})$ with $g\in C^{\infty}(\mathbb{H})$ :

	$\displaystyle d^{t}(g\,(X^{\ast}\wedge Y^{\ast}))=$	$\displaystyle\star d(g\,T^{\ast})=\star(-g\,(X^{\ast}\wedge Y^{\ast})+Xg\,(X^{\ast}\wedge T^{\ast})+Yg\,(Y^{\ast}\wedge T^{\ast}))$
	$\displaystyle=$	$\displaystyle-g\,T^{\ast}-Xg\,Y^{\ast}+Yg\,X^{\ast}\in\Omega^{1,\geq 1}(\mathbb{H})$

and so $d^{t}(\Omega^{2,\geq 2}(\mathbb{H}))\subset\Omega^{1,\geq 1}(\mathbb{H})$ , but not $\Omega^{1,\geq 2}(\mathbb{H})$ .

Applying similar computations, one can easily check that the transpose of the operator ${\rm gr}(d)^{\Phi}$ instead respects the filtration.

A situation similar to Example B.1 also holds for the algebraic part of the de Rham differential $\tilde{d}\colon\Omega^{\bullet}(\mathbb{G})\to\Omega^{\bullet}(\mathbb{G})$ , when considering a nilpotent Lie group $\mathbb{G}$ with a filtration on its Lie algebra $\mathfrak{g}$ that is not coming from a homogeneous structure as in the following example:

Example B.2.

Let us consider the 4-dimensional Engel group $\mathbb{G}$ , that is, the connected simply connected nilpotent Lie group with Lie algebra $\mathfrak{g}={\rm span}_{\mathbb{R}}\{X_{1},X_{2},X_{3},X_{4}\}$ and $[X_{1},X_{i}]=X_{i+1}$ for $i=2,3$ . We identify $\mathfrak{g}$ with the space of left-invariant vector fields, and we consider the following left-invariant filtration of $T\mathbb{G}$

\displaystyle\{0\}=H^{0}\subset H^{1}={\rm span}_{\mathbb{R}}\{X_{1},X_{2}\}\subset H^{2}={\rm span}_{\mathbb{R}}\{X_{1},X_{2},X_{3},X_{4}\}=T\mathbb{G}\,.

The coframe of the canonical frame $\mathbb{X}=(X_{1},,X_{2},X_{3},X_{4})$ is denoted by $\Theta=(\theta^{1},\theta^{2},\theta^{3},\theta^{4})$ Then, for an arbitrary form $f_{1}\,\theta^{3}+f_{2}\,\theta^{4}$ in $\Omega^{1,\geq 2}(\mathbb{G})$ , with $f_{1},f_{2}\in C^{\infty}(\mathbb{G})$ , we have

\displaystyle\tilde{d}(f_{1}\,\theta^{3}+f_{2}\,\theta^{4})=

\displaystyle-f_{1}\,(\theta^{1}\wedge\theta^{2})-f_{2}\,(\theta^{1}\wedge\theta^{3})\ \in\Omega^{2,\geq 2}(\mathbb{G})\,.

However, given a form $g\,(\theta^{1}\wedge\theta^{3})\ \in\Omega^{2,\geq 3}(\mathbb{G})$ with $g\in C^{\infty}(\mathbb{G})$ , we obtain $\tilde{d}^{t}(g\,(\theta^{1}\wedge\theta^{3}))=-g\,\theta^{4}\in\Omega^{1,\geq 2}(\mathbb{G})$ .

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Subcomplexes on filtered Riemannian manifolds

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

1.1. Motivation

1.2. Sketch of the constructions of the complexes

1.2.1. Constructing DD using d0td_{0}^{t}

1.2.2. Constructing DD using d0−1d_{0}^{-1}

1.2.3. Constructions of D~\tilde{D}

1.3. Novelty and future works

Example 1.1.

Example 1.2.

1.4. Organisation of the paper

1.5. Acknowledgements

2. Preliminaries

2.1. Filtrations of vector spaces

2.1.1. Filtered vector spaces

2.1.2. The vector space gr​(W){\rm gr}(W)

2.1.3. The linear map gr​(T){\rm gr}(T)

2.1.4. The basis ⟨e⟩\langle e\rangle

2.1.5. Matrix representation of linear maps

2.1.6. Subfiltration

2.1.7. Decreasing labelling and duality

2.1.8. Graded vector spaces

Definition 2.1.

2.1.9. Filtration on the exterior algebra.

Lemma 2.2.

2.1.10. Euclidean filtered vector spaces

2.2. Preliminaries on manifolds

2.2.1. The maps dd and d~\tilde{d} on a general manifold

Lemma 2.3.

Sketch of the proof.

2.3. Some natural concepts

2.3.1. Frame and coframe

2.3.2. Algebraic maps on bundles

Remark 2.4.

2.3.3. Metric on bundles

2.4. Filtered tangent bundles

2.4.1. Adapted frames

Definition 2.5.

Definition 2.6.

Definition 2.7.

Definition 2.8.

2.4.2. Weights of forms

Example 2.9.

Example 2.10.

Definition 2.11.

2.4.3. The bundle ∧kgr​(T∗​M)\wedge^{k}{\rm gr}(T^{*}M) and its weights

Example 2.12.

Example 2.13.

Lemma 2.14.

2.4.4. The gr{\rm gr} operation

Definition 2.15.

Lemma 2.16.

Definition 2.17.

2.5. Tangent bundles equipped with a filtration and a metric

2.5.1. Identification of Ω∙​(M)\Omega^{\bullet}(M) and Γ​(∧∙gr​(T∗​M))\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))

Lemma 2.18.

Lemma 2.19.

Proof.

2.5.2. Scalar products on ∧∙Tx∗​M\wedge^{\bullet}T_{x}^{*}M and of ∧∙gr​(Tx∗​M)\wedge^{\bullet}{\rm gr}(T_{x}^{*}M)

2.5.3. Hodge-star, scalar product on Ωc∙​(M)\Omega_{c}^{\bullet}(M) and Γc​(∧∙gr​(T∗​M))\Gamma_{c}(\wedge^{\bullet}{\rm gr}(T^{*}M)), and transpose

2.5.4. Properties of the transpose for the L2L^{2}-inner product

Lemma 2.20.

Proof.

2.5.5. Properties of gr{\rm gr} for algebraic maps

Lemma 2.21.

Proposition 2.22.

Proof.

Corollary 2.23.

3. Filtered manifolds and osculating objects

3.1. Setting and definitions

Example 3.1.

3.1.1. Bundle of osculating Lie groups and algebras

Definition 3.2.

3.1.2. The osculating Chevalley-Eilenberg differential

3.1.3. The osculating box operator

3.1.4. The partial inverse d𝔤​M−1d^{-1}_{\mathfrak{g}M}

3.2. The maps dd and d~\tilde{d} on a filtered manifold

1.2.1. Constructing $D$ using $d_{0}^{t}$

1.2.2. Constructing $D$ using $d_{0}^{-1}$

1.2.3. Constructions of $\tilde{D}$

2.1.2. The vector space ${\rm gr}(W)$

2.1.3. The linear map ${\rm gr}(T)$

2.1.4. The basis $\langle e\rangle$

2.2.1. The maps $d$ and $\tilde{d}$ on a general manifold

2.4.3. The bundle $\wedge^{k}{\rm gr}(T^{*}M)$ and its weights

2.4.4. The ${\rm gr}$ operation

2.5.1. Identification of $\Omega^{\bullet}(M)$ and $\Gamma(\wedge^{\bullet}{\rm gr}(T^{*}M))$

2.5.2. Scalar products on $\wedge^{\bullet}T_{x}^{}M$ and of $\wedge^{\bullet}{\rm gr}(T_{x}^{}M)$

2.5.3. Hodge-star, scalar product on $\Omega_{c}^{\bullet}(M)$ and $\Gamma_{c}(\wedge^{\bullet}{\rm gr}(T^{*}M))$ , and transpose

2.5.4. Properties of the transpose for the $L^{2}$ -inner product

2.5.5. Properties of ${\rm gr}$ for algebraic maps

3.1.4. The partial inverse $d^{-1}_{\mathfrak{g}M}$

3.2. The maps $d$ and $\tilde{d}$ on a filtered manifold

4.1. Base differential and codifferential $d_{0}$ and $\delta_{0}$

4.2. The operators $\Box$ , $P$ and $L$

4.3. The complexes $(E_{0}^{\bullet},D)$ and $(F_{0}^{\bullet},C)$

4.4. The complexes $(E_{0}^{\bullet},\tilde{D})$ and $(F_{0}^{\bullet},\tilde{C})$

4.5. Equivalent constructions for the subcomplexes $(E_{0}^{\bullet},D)$ and $(E_{0}^{\bullet},\tilde{D})$

Description of $\tilde{d}$

Description of $d$

Description of $d_{\mathfrak{g}M}$

Description of $d_{0}$ and $d_{0}^{t}$

Description of $\Box_{0}$ , $P_{0}$

Description of $\Box$ , $P$ , $L$

Computation of $D$

Description of $\tilde{\Box}$ , $\tilde{P}$ , $\tilde{L}$

Computation of $\tilde{D}$