Global Hypersurfaces of Section for Geodesic Flows on Convex Hypersurfaces

Sunghae Cho and Dongho Lee were supported by National Science Foundation of Korea Grant NRF2023005562 funded by the Korean Government.

Let $Y$ be a smooth closed manifold, i.e.  a compact manifold without boundary, and $X$ be a non-vanishing vector field on $Y$. A global hypersurface of section for $X$ is an embedded submanifold $P\subset Y$ of codimension 1 with (possibly empty) boundary $\partial P=B$ such that

• $\cdot$

  the vector field $X$ is transverse to the interior $\mathring{P}$ of $P$,

• $\cdot$

  the boundary $B$ is $X$-invariant; in other words, $X$ is tangent to $B$,

• $\cdot$

  for any $p\in Y$, there exist $t_{+},t_{-}>0$ such that $Fl^{X}_{t_{+}}(p),Fl^{X}_{-t_{-}}(p)\in P$.

If $P$ is a global hypersurface of section, we can define the (first) return time $\tau_{p}$ for each $p\in\mathring{P}$ by $\tau_{p}=\min\{t>0:Fl^{X}_{t}(p)\in P\}$ and the (first) return map by $\Psi(p)=Fl^{X}_{\tau_{p}}(p)$.

Open book decompositions are closely related to the global hypersurfaces of section. An open book decomposition on a closed manifold $Y$ is a pair $(B,\pi)$ of a codimension 2 closed submanifold $B$ and a map $\pi$ from $Y\setminus B$ to $S^{1}\subset\mathbb{C}$ which satisfies the following.

• $\cdot$

  The normal bundle of $B$ is trivial. We call $B$ the binding. Let $\xi:B\times D^{2}\to\nu(B)$ be a fixed trivialization of the normal bundle, which is identified with a tubular beighborhood, of $B$.

• $\cdot$

  The map $\pi$ is a fiber bundle such that $(\pi\circ\xi)(b;r,\theta)=e^{i\theta}$ on $\nu(B)\setminus B$, where $(r,\theta)$ is a polar coordinate on $D^{2}$. We call the closure of each fiber $\overline{\pi^{-1}(e^{i\theta})}=P_{\theta}$ the page. Note that $\partial P_{\theta}=B$ for any $\theta$.

Let $X$ be a vector field on $Y$, and $(B,\pi)$ be an open book on $Y$. If $X$ is transverse to each page $P_{\theta}$ and tangent to $B$, then we say $X$ is adapted to $(B,\pi)$.

If a vector field $X$ is adapted to $(B,\pi)$, then each page $P_{\theta}=\pi^{-1}(e^{i\theta})$ can be regarded as a candidate for the global hypersurface of section. There might exist an orbit of $X$ which does not return to the page in a finite time. Such an orbit should be asymptotic to the boundary as $t$ becomes large. A discussion about a case of dimension 3 can be found in [HWZ98]. In 3.5, we will show that the case of unbounded return time does not appear in our setting.

Let $W$ be a manifold without boundary and $\omega$ be a 2-form on $W$. If $\omega$ is a non-degenerate closed form, we say $\omega$ is a symplectic form and $(W,\omega)$ is a symplectic manifold. A symplectomorphism is a diffeomorphism between symplectic manifolds which preserves the symplectic form. Let $W^{\prime}\subset W$ be a submanifold such that $\omega|_{W^{\prime}}$ is also a symplectic form on $W^{\prime}$. We call $W^{\prime}$ a symplectic submanifold. A diffeomorphism between manifolds induces a symplectomorphism between cotangent bundles via pullback. Also, if $N$ is a submanifold of $M$, then $T^{*}N$ is a symplectic submanifold of $T^{*}M$.

Let $(W,\omega)$ be a symplectic manifold with a boundary, and assume that $\omega$ has a primitive $\lambda$, i.e.  $\omega=d\lambda$. A vector field $X$ such that $i_{X}\omega=\lambda$ is called Liouville vector field. If a Liouville vector field $X$ exists and defines an outward vector field on the boundary, we call $(W,\lambda)$ a Liouville domain.

Given a smooth function $H:W\to\mathbb{R}$, we can associate a vector field $X_{H}$ to $H$ via

$$\left(i_{X_{H}}\right)\omega(-)=\omega(X_{H},-)=dH(-).$$

This is well-defined by non-degeneracy of $\omega$. We say $X_{H}$ is a Hamiltonian vector field, and in this sense, we call $H$ a Hamiltonian function or simply Hamiltonian. If a diffeomorphism can be written as a time 1-flow of a Hamiltonian vector field, we say it is a Hamiltonian diffeomorphism. Note that we can also use time-dependent Hamiltonian $H:W\times\mathbb{R}\to\mathbb{R}$ and get time-dependent Hamiltonian diffeomorphism. By definition, a Hamiltonian vector field $X_{H}$ vanishes at a point $p$ if and only if $d_{p}H=0$. A Hamiltonian diffeomorphism is a symplectomorphism, as $\left(Fl^{X_{H}}\right)^{*}\omega=\mathcal{L}_{X_{H}}\omega=di_{X_{H}}\omega=0$ by the Cartan magic formula. An important observation is that the Hamiltonian vector field of an autonomous Hamiltonian is tangent to the regular level set of the generating Hamiltonian. In other words, the value of $H$ is preserved under the flow of $X_{H}$. More detailed explanations with examples about symplectic geometry and Hamiltonian mechanics can be found in [Arn89], [CdS01], [Ber01], [HZ11] or [MS17].

Let $H,F:W\to\mathbb{R}$ be Hamiltonians. The Poisson bracket is defined by $\{H,F\}:=\omega(X_{H},X_{F})$. It’s clear that the Poisson bracket is alternating. If $(W,\omega)=(T^{*}\mathbb{R}^{n},\omega_{\mathrm{std}})$, we have the following formula, which can be found for example in the chapter 1 of [MS17]

$$\{H,F\}=\sum_{j}\frac{\partial H}{\partial y_{j}}\frac{\partial F}{\partial x_{j}}-\frac{\partial H}{\partial x_{j}}\frac{\partial F}{\partial x_{j}}.$$

Let $Y$ be a $(2n+1)$-dimensional manifold, and $\xi$ be a $2n$-dimensional distribution on $Y$. Then we can locally write $\xi$ as $\ker\alpha$ for some 1-form $\alpha$. Assume that $\xi$ is coorientable, i.e.  $TW/\xi$ is orientable. Then we can find a globally defined 1-form $\alpha$. If $\alpha\wedge(d\alpha)^{n}$ is a volume form, we say $\alpha$ is a contact form, $\xi=\ker\alpha$ is a contact structure, and $(W,\xi)$ is a contact manifold. A contact form $\alpha$ defines a unique vector field $R$ such that $\alpha(R)=1$, $i_{R}d\alpha=0$. We call $R$ the Reeb vector field. Note that the Reeb vector field is always non-vanishing. A standard example of a contact manifold is the boundary of a Liouville domain $(W,\lambda)$ with contact form $\lambda|_{\partial W}$, and a regular level set of Hamiltonian function $H:(W,d\lambda)\to\mathbb{R}$ with the same contact form. In the second case, the Hamiltonian vector field $X_{H}$ restricted to the regular level set of $H$ is the Reeb vector field. We can consider an open book decomposition $(B,\pi)$ on a contact manifold $(Y,\ker\alpha)$ which the Reeb vector field adapted to. The close relationship between contact structures on a manifold and an open book decomposition is explored in [Gir02].

Let $(M,g)$ be a complete Riemannian manifold. For each $x\in M$ and $v\in T_{p}M$, we have a unique geodesic $\gamma_{x,v}$ with the initial condition $\gamma_{x,v}(0)=x$, $\dot{\gamma}_{x,v}(0)=v$. The geodesic flow is a 1-parameter family of diffeomorphisms on $TM$ defined by $\Phi_{t}(x,v)=(\gamma_{x,v}(t),\dot{\gamma}_{x,v}(t))$. By differentiating $\Phi_{t}$ by $t$, we get the geodesic vector field on $TM$ which generates the geodesic flow.

The metric $g$ on $TM$ induces the dual metric $g^{*}$ on $T^{*}M$ by natural pairing, and we can also define the (co-)geodesic flow and (co-)geodesic vector field on $T^{*}M$.

The regular level set $H^{-1}(0)$ is a unit cotangent bundle $ST^{*}M$, which naturally is a contact manifold, whose Reeb vector field is Hamiltonian vector field $X_{H}$ restricted to $ST^{*}M$.

For notational convenience, we write a point $(x_{1},\ldots,x_{n})$ in $\mathbb{R}^{n}$ by $\vec{x}$ so that $x=(x_{0},\vec{x})$ is a point in $\mathbb{R}^{n+1}$. For a function $f:\mathbb{R}^{n+1}\to\mathbb{R}$, we denote the partial derivative $\partial_{x_{i}}f$ by $f_{i}$, and the gradient vector field $(f_{0},\ldots,f_{n})$ by $\nabla f$. We will also write the Hessian matrix of $f$ which is computed in $\mathbb{R}^{n+1}$ by $\text{Hess}(f)=(\partial_{x_{i}}\partial_{x_{j}}f)_{ij}=(f_{ij})$. We use coordinate $(y_{0},\cdots,y_{n})$ for the cotangent fiber $T^{*}_{x}\mathbb{R}^{n+1}$, and also for the cotangent fiber of a hypersurface contained in $\mathbb{R}^{n+1}$.

Let $f:\mathbb{R}^{n+1}\to\mathbb{R}$ be a smooth function, and $0$ be a regular value. The level set $M=f^{-1}(0)$ is an $n$-dimensional Riemannian manifold, whose metric is inherited from $\mathbb{R}^{n+1}$. We can embed $T^{*}M$ into $T^{*}\mathbb{R}^{n+1}$ by

$$T^{*}M=\left\{(x,y)\in T^{*}\mathbb{R}^{n+1}:f(x)=0,\,\,y\cdot\nabla f=0\right\}.$$

Here, $\cdot$ is a standard inner product on $\mathbb{R}^{n+1}$, and we identified $T^{*}M$ to $TM$ by the metric on $M$. Let $\tilde{H}=\frac{1}{2}\left(||y||^{2}-1\right)$ on $T^{*}\mathbb{R}^{n+1}$, and $H=\tilde{H}|_{T^{*}W}$. Define $\tilde{f},g:T^{*}\mathbb{R}^{n+1}\to\mathbb{R}$ by

$$\tilde{f}(x,y)=f(x),\quad g(x,y)=y\cdot\nabla f$$

so that $T^{*}M$ is an intersection $\tilde{f}^{-1}(0)\cap g^{-1}(0)$.

From a straightforward computation, we have

$$X_{\tilde{H}}=\sum_{j}y_{j}\frac{\partial}{\partial x_{j}},\quad X_{\tilde{f}}=-\sum_{j}\frac{\partial f}{\partial x_{j}}\frac{\partial}{\partial y_{j}}.$$

Using the formula for the Poisson bracket, we have

	$\displaystyle\{\tilde{f},g\}$	$\displaystyle=\sum\frac{\partial f}{\partial x_{j}}\frac{\partial g}{\partial y_{j}}=\sum\left(\frac{\partial f}{\partial x_{j}}\right)^{2}=||\nabla f||^{2},$
	$\displaystyle\{\tilde{f},\tilde{H}\}$	$\displaystyle=\sum\frac{\partial f}{\partial x_{j}}\frac{\partial\tilde{H}}{\partial y_{j}}=\sum\frac{\partial f}{\partial x_{j}}y_{j}=y\cdot\nabla f=0,$
	$\displaystyle\{g,\tilde{H}\}$	$\displaystyle=\sum\frac{\partial g}{\partial x_{j}}\frac{\partial\tilde{H}}{\partial y_{j}}=\sum_{j}\frac{\partial}{\partial x_{j}}\left(\sum_{i}y_{i}\frac{\partial f}{\partial x_{i}}\right)y_{j}=\sum_{i,j}\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}y_{i}y_{j}=\mathrm{Hess}(f)(y,y).$

To sum up, we have the following.

Alternatively, this formula can be derived by orthogonal projection.

An important quantity associated to the Riemannian manifold $(M,g)$ is the sectional curvature $K_{M}(-,-)$, which is defined for a point $x\in M$ and two vectors $v,w$ in $T_{p}M$. This can be regarded as the Gauss curvature of a 2-dimensional subspace ‘spanned’ by $v,w$. For the precise definition and properties of sectional curvature, we refer to [Mil63], [KN63], [Spi79], or [dC92]. Let $N\subset M$ be an oriented hypersurface, and $\nu$ be the unit normal vector field on $N$.

The formula becomes simpler in the case of a hypersurface in the Euclidean space, whose sectional curvature vanishes.

We consider the case $f:\mathbb{R}^{n+1}\to\mathbb{R}$ is a smooth function and $0$ is a regular value. Let $M$ be the regular level set $f^{-1}(0)$. We assume the following conditions for $f$.

1. (A1)

   For any point $(x_{0},\vec{x})$ in $M$ such that $|x_{0}|$ is small enough, $f(x_{0},\vec{x})=f(-x_{0},\vec{x})$.

2. (A2)

   The Hessian of $f$ is positive definite. Explicitly, for any $(x,y)\in T^{*}M$, $y\neq 0$, $\mathrm{Hess}(f)_{x}(y,y)>0$.

Here is a classical theorem which explains the role of (A1). For the proof, see [KN69] Chapter VII.8.

Let $N$ be the codimension $1$ submanifold $N=M\cap\{(x,y):x_{0}=0\}.$ Then it is clear that $N$ is a fixed point set of a locally defined isometric involution $i_{M}(x_{0},\vec{x})=(-x_{0},\vec{x})$, and it follows that $N$ is a totally geodesic submanifold from 3.1. Let $Y=ST^{*}M$ and $B=ST^{*}N$. Then $B$ can be written as

$$B=\{(x,y)\in ST^{*}M:x_{0}=0,y_{0}=0\}.$$

We note the triviality of the normal bundle of $B$ in $Y$.

Condition (A2) implies that $f$ is a convex function, and $M$ bounds a compact convex domain. It follows that $M$ is diffeomorphic to $S^{n}$, where the diffeomorphism can be explicitly written as $x\mapsto x/||x||$, assuming that $0$ is in the interior of the bounding region of $M$. Since $M$ is convex and $0$ is in the interior of the domain $M$ bounds, this is well-defined.

With the setting of the previous section, we define a map $\pi:Y\setminus B\to S^{1}\subset\mathbb{C}$ by

$$\pi(x,y)=\frac{x_{0}+iy_{0}}{|x_{0}+iy_{0}|}.$$

The angular form is defined by

$$\Theta=i\cdot d\log\pi=\frac{y_{0}dx_{0}-x_{0}dy_{0}}{x_{0}^{2}+y_{0}^{2}}=\frac{\theta}{x_{0}^{2}+y_{0}^{2}}.$$

Put $X_{H}$ which was computed in 2.6 into $\theta$, we have the following.

$$\theta(X_{H})=y_{0}^{2}+x_{0}^{2}\frac{\mathrm{Hess}(f)_{x}(y,y)}{\|\nabla f(x)\|^{2}}\frac{f_{0}(x)}{x_{0}}.$$

We define a function $A=A(x,y)$ by

$$A(x,y)=\frac{\mathrm{Hess}(f)_{x}(y,y)}{\|\nabla f(x)\|^{2}}\frac{f_{0}(x)}{x_{0}}$$

so that $\theta(X_{H})=A(x,y)x_{0}^{2}+y_{0}^{2}$.

Note that if we take $n=2$, then $M$ is 2-sphere so $P=P_{\pi/2}$ in 3.5 is the Birkhoff annulus. In this sense, our construction can be regarded as a generalization of the Birkhoff annulus.

We can apply 2.8 on our hypersurface $M$. The unit normal vector $\nu$ is $\frac{\nabla f}{||\nabla f||}$, and we have

$$S\left(\frac{\partial}{\partial x_{i}}\right)=\frac{\partial}{\partial x_{i}}\frac{\nabla f}{||\nabla f||}=\sum_{j}\left(\frac{f_{ij}}{||\nabla f||}-\frac{\sum_{k}f_{k}f_{ki}f_{j}}{||\nabla f||^{3}}\right)\frac{\partial}{\partial x_{j}}.$$

It follows that

$$S(v,w)=\frac{\sum_{i,j}v_{i}f_{ij}w_{i}}{||\nabla f||}-\frac{\sum_{i,j,k}v_{i}f_{ik}f_{k}f_{j}w_{j}}{||\nabla f||^{3}}=\frac{\mathrm{Hess}(f)(v,w)}{||\nabla f||}$$

for $v,w\in TM$. The second term vanishes because $w\in TM$ implies that $w\cdot\nabla f=\sum_{j}w_{j}f_{j}=0$.

Put this into 2.8, and we get the following.

Hence, the sign of $K_{M}$ depends on the sign of the ($2\times 2$)-minors of $\mathrm{Hess}(f)$. Since the formula holds for any $v$ and $w$, we have the following.

The following corollary can also be found in various literature, for example [Sac60].

Now we can formulate 3.5 in another form.

Using the same notations from the previous sections, we will explore the topology of $N$ and $P$.