Killing vector fields on Riemannian and Lorentzian 3-manifolds

The aim of this paper is to give a complete local classification of all Riemannian 3-manifolds $(M,g)$ that admit a nonvanishing Killing vector field $T$. In fact this classification will also yield a related one: that of all Lorentzian 3-manifolds supporting a timelike Killing vector field. Our classification proceeds by considering the special case when $T$ has constant length: the general case follows from this one by applying a conformal change by the factor of $g(T,T$). But in fact there are important reasons for imposing this condition. One of them is that constant length allows us to adapt the machinery of the Newman-Penrose formalism [NP62] — a construct that originated in 4-dimensional Lorentzian geometry — to the setting of 3-dimensional Riemannian geometry; see also [SW14, NTC15, BS18], wherein similar frame techniques have been applied in dimension 3.

As shown in Section 2 below, constant length is a prerequisite for this formalism. But more importantly, our interest in constant length arises from what we regard as the “canonical” constant length Killing vector field in dimension 3: the unit length Killing vector field $T$ on $(\mathbb{S}^{3},\mathring{g})$ tangent to the Hopf fibration, where $\mathring{g}$ is the standard (round) metric. Given the special geometry of $(\mathbb{S}^{3},\mathring{g})$ as a spherical space form, and the presence of such a vector field on it, we take our motivation from the following questions:

1. 1.

   Can one classify locally all Riemannian 3-manifolds admitting a constant length Killing vector field? Do they take on a “canonical” form?

2. 2.

   If a Riemannian 3-manifold admits a constant length Killing vector field, then when will it be locally conformally flat, as with $(\mathbb{S}^{3},\mathring{g})$?

(Yet another path of inquiry, which we do not pursue here, would be to examine when the circle action provided by a constant length Killing vector field is free, and the role that sectional curvature plays in this; see [BN08].) A complete answer to our first question above is provided in our first Theorem:

The following remarks help to shed light on this result:

1. i.

   After our first preprint appeared, we learned of the works [Man14, LM17], in which the existence of coordinates isometric to (1) are proved, as well as a result that includes (2) as a special case; these were obtained via a different method than ours, and applied to the classification of Riemannian submersions from 3-manifolds to a surface, whose fibers are the integral curves of a Killing vector field.

2. ii.

   An almost identical Theorem exists for unit timelike Killing vector fields on Lorentzian 3-manifolds; see Corollary 2 in Section 6 below, wherein the relevant Lorentzian terminology is also defined.

3. iii.

   If $T$ does not have unit length, then (1) is scaled by $g(T,T)$. Also, in the global case $M=\mathbb{R}^{3}$, completeness can be obtained by conditions other than $\text{Ric}(T,T)=0$, using, e.g., [MR20, Theorem 6].

4. iv.

   The “canonical form” alluded to above is manifested in (1) and (2); for the form of the metric in the coordinate basis $\{\partial_{t},\partial_{r},\partial_{\theta}\}$, see (36) in Section 5 below. As Theorem 1 makes clear, our classification depends entirely on two functions, the scalar curvature $S$ and $\text{Ric}(T,T)$. Let us say more about the latter function, which is especially important; one way to appreciate its significance when $\text{dim}\,M=3$ is as follows. If a vector field $T$ has constant length and geodesic flow (as does any constant length Killing vector field), then the function $\text{Ric}(T,T)$, if nonnegative, completely governs whether its orthogonal complement $T^{\perp}\subseteq TM$ is integrable. As a consequence, it was shown in [HP16] that when such a $T$ satisfies $\text{Ric}(T,T)>0$ and when $M$ is orientable and compact, then $T^{\flat}$ is a contact form and $T$ is its Reeb vector field; if in addition $T$ is divergence-free and $\text{Ric}(T,T)=1$, then $T_{*}(T^{\perp})$ is $J$-invariant, where $J$ is the Levi-Civita almost-complex structure on $TTM$ (in fact these two conditions are necessary and sufficient).

5. v.

   The previous remark did not assume that $T$ is a unit length Killing vector field. Imposing this condition — as well as the condition $\text{Ric}(T,T)=2$, so that the endomorphism in (5) below defines an almost complex structure on $T^{\perp}$ — would make $(M,g,T)$ a Sasakian structure. In dimension 3, a classification of these on closed manifolds was obtained in [Gei97], up to diffeomorphism; an explicit metric classification was then given in [Bel01, Bel03], which also established a one-to-one correspondence between Sasakian and normal CR structures, and also classified the latter. For an application to monopole fields, see [DH19].

Finally, our answer to the second question is also given in terms of $\text{Ric}(T,T)$:

As a check of the last statement, note that for $(\mathbb{S}^{3},\mathring{g})$ with Hopf Killing vector field $T$ and radius $R$,

A Killing vector field $T$ on a Riemannian manifold $(M,g)$ is defined by the condition $\mathfrak{L}_{T}g=0,$ where $\mathfrak{L}$ is the Lie derivative. However, when $T$ has unit length and when $\text{dim}\,M=3$, there is an equivalent formulation, given by Lemma 2 below, which plays a crucial role in our classification. This formulation also involves the Lie derivative, but owing to the low dimension, only certain components of it, which components carry geometric properties of the flow of $T$. These properties are the divergence, twist, and shear; as the latter two are not as well known as the former, we now digress to define them explicitly. Thus, let $T$ be a smooth unit length vector field defined in an open subset of a Riemannian 3-manifold $(M,g)$, so that $\nabla_{\!v}{T}\perp T$ for all vectors $v$ ($\nabla$ is the Levi-Civita connection). Let $X$ and $Y$ be two smooth vector fields such that $\{T,X,Y\}$ is a local orthonormal frame. Now define the following endomorphism $D$ of the orthogonal complement $T^{\perp}\subseteq TM$,

and observe that its matrix with respect to the frame $\{T,X,Y\}$ is

Contained within this matrix are three geometric properties associated to the flow of $T$:

1. 1.

   The divergence of $T$, denoted $\text{div}\,T$, is simply the trace of $D$.

2. 2.

   By Frobenius’s theorem, $T^{\perp}$ is integrable if and only if the anti-symmetric part of $D$ vanishes; as seen in (8) below, this vanishing is governed by the following function, which comprises the off-diagonal elements of the anti-symmetric part of $D$:

   $\displaystyle\omega\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=g({T},{[X,Y]})=g({\nabla_{\!Y}{T}},{X})-g({\nabla_{\!X}{T}},{Y}).$

   (6)

   Since $\omega^{2}$ equals the determinant of the anti-symmetric part of $D$, it is a frame independent quantity. We call $\omega^{2}$ the twist function of $T$ and say that the flow of $T$ is twist-free if $\omega^{2}=0$.

3. 3.

   The third piece of information is the shear $\sigma$ of $T$; it is given by the trace-free symmetric part of $D$, whose components $\sigma_{1},\sigma_{2}$ we combine here into a complex-valued quantity:

   $\displaystyle\hskip 18.7898pt\sigma\!\!$

   $\displaystyle\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=$

   $\displaystyle\!\!\underbrace{\,\frac{1}{2}\Big{(}g({\nabla_{\!Y}{T}},{Y})-g({\nabla_{\!X}{T}},{X})\Big{)}\,}_{\sigma_{1}}+\ i\underbrace{\,\frac{1}{2}\Big{(}g({\nabla_{\!Y}{T}},{X})+g({\nabla_{\!X}{T}},{Y})\Big{)}\,}_{\sigma_{2}}.$

   Although $\sigma$ itself is not frame independent, its magnitude $|\sigma^{2}|$ is: by (8) below, it is minus the determinant of the trace-free symmetric part of $D$. We say that the flow of $T$ is shear-free if $\sigma=0$. As with being twist-free, being shear-free is a frame independent statement. In terms of $\text{div}\,T$, $\omega$, and $\sigma$, $D$ takes the form

   $\displaystyle D=\begin{pmatrix}\frac{1}{2}\text{div}\,T&0\\ 0&\frac{1}{2}\text{div}\,T\end{pmatrix}\ +\!\!\!\underbrace{\,\begin{pmatrix}-\sigma_{1}&\sigma_{2}\\ \sigma_{2}&\sigma_{1}\end{pmatrix}\,}_{\text{trace-free symmetric}}\!\!\!+\,\underbrace{\,\begin{pmatrix}0&\frac{\omega}{2}\\ -\frac{\omega}{2}&0\end{pmatrix}\,}_{\text{anti-symmetric}}\cdot$

   (8)

We record here the well known fact that, while being divergence-free is not a conformal invariant, being shear-free or twist-free is:

When $\text{dim}\,M\geq 4$, divergence and shear alone are not enough to characterize unit length Killing vector fields, but they do when $\text{dim}\,M=3$:

We can now state our plan of attack: divergence, geodesic flow, twist, and shear all involve first derivatives of $T$, whereas curvature involves second derivatives. Our plan of attack, therefore, is to express the components of the Riemann curvature tensor in terms of the divergence, twist, and shear of $T$, thereby reducing second-order equations to first-order ones — indeed, further encouraged by the fact that, as we have just seen, if $T$ is a unit length Killing vector field, then $\text{div}\,T,\sigma$, and $\nabla_{\!T}{T}$ all vanish, so that only $T$’s twist function $\omega^{2}$ is unknown. The hope is that this will simplify things enough to allow a full determination of the metric. And it will — after we express the curvature in terms of the divergence, twist, and shear, which we now proceed to do.

In what follows we present the Newman-Penrose formalism for Riemannian 3-manifolds, presenting here only the resulting equations; complete derivations can be found in [Aaz15]. Let $\{T,X,Y\}$ be an orthonormal frame — with $T$ not necessarily a Killing vector field — and form the complex-valued quantities

Henceforth we work with the complex frame $\{T,{\boldsymbol{m}},\overline{\boldsymbol{m}}\}$, for which only

are nonzero. The following quantities associated to this complex frame play a central role in all that follows.

Note that, because $T$ has unit length, its flow is geodesic, $\nabla_{\!T}{T}=0$, if and only if $\kappa=0$; that $\sigma$, when written out in terms of its real and imaginary parts, is precisely the complex shear (LABEL:eqn:shear); and that the spin coefficient $\rho$ has real and imaginary parts given by

In other words, the first three spin coefficients $\kappa,\rho,\sigma$ stand in for the geometric properties of the flow of $T$ discussed above. In terms of all five spin coefficients, the Lie brackets are

(The remaining Lie bracket $[T,\overline{\boldsymbol{m}}]$ is obtained by complex conjugation.) Now to the curvature; to begin with, our sign convention for the Riemann curvature tensor is

in which case the Ricci tensor with respect to the complex frame $\{T,{\boldsymbol{m}},\overline{\boldsymbol{m}}\}$ is

The following identities satisfied by the Ricci tensor in the complex frame $\{T,{\boldsymbol{m}},\overline{\boldsymbol{m}}\}$ will appear in formulae below:

The Newman-Penrose begins by expressing the Lie brackets in terms of spin coefficients, as we saw in (11) and (12) above. It then moves down to the level of curvature, by expressing the following components of the curvature tensor,

in terms of the Ricci tensor and the spin coefficients. Doing so, the following (first-order) equations arise; they play the driving role in our classification:

Finally, up to complex conjugation, there are two nontrivial differential Bianchi identities:

We now immediately specialize to the case when $T$ is a Killing vector field:

We have not yet considered the differential Bianchi identities; let us do so now. Inserting the contents of Lemma 3 into (3) and (3), as well as $\bar{\rho}=-\rho$, yields

for (3); but this is precisely the Lie bracket (11) applied to $\rho$ (bearing in mind that $T(\rho)=0$), and therefore carries no new information. As for (3), it yields

where $T(S)=0$ and $\bar{\rho}=-\rho$ have been used. But this is precisely the Lie bracket (12) applied to $\rho$, so that (3) also yields no new information.

The goal of this section is to establish the “right” local coordinates in which to prove Theorem 1 in the next section. To begin with, recall that because

the only spin coefficients remaining are $\varepsilon$ and $\beta$. Observe that the former is in fact purely imaginary,

and the latter, when $\nabla_{\!T}{T}=0$, is given by (23). The following “gauge freedom” simultaneously enjoyed by these two spin coefficients will prove useful in the proof of Theorem 1:

The following Corollary collects together what we’ve established so far:

Notice that (31) implies that such a frame may not always exist globally; e.g., if $M$ is compact and $S$ is nonnegative and positive somewhere, then a standard Riccati argument yields that in such a case the only complete solution to (31) is one where $\text{div}\,Y=S+\frac{\omega^{2}}{2}=0$, which is impossible. We now proceed to our local classification.

Theorem 1 follows from one further modification to the orthonormal frame satisfying (30):