Geodesic flows: notes 7

Geodesic flows, notes 7: recap

The section numbers correspond to the numbering of previous blog entries. $ \newcommand{\vali}{(-\varepsilon,\varepsilon)} \newcommand{\id}{\operatorname{id}} \newcommand{\im}{\operatorname{im}} \newcommand{\Der}[1]{\frac{d}{d#1}} \newcommand{\Sa}{{\text{Sasaki}}} \newcommand{\ip}[2]{\left\langle#1,#2\right\rangle} \newcommand{\R}{{\mathbb R}} $

0. The goal

The goal is to show that a function on a suitable Riemannian manifold $(M,g)$ is uniquely determined by its integrals over all maximal geodesics. To this end, we must understand the geometry of geodesics in great detail. The geodesic X-ray transform takes a function $f\colon M\to\R$ to a function $If\colon\Gamma\to\R$, where $\Gamma$ is the set of all geodesics on $M$. This $I$ is a linear integral transform, and it is known as the geodesic X-ray transform. The question is whether this transform is injective.

1. Symplectic manifolds and Hamilton flows

Definition. A symplectic manifold is a pair \((N, \sigma)\) where \(N\) is an \(2n\)-dimensional smooth manifold, and \(\sigma\) is symplectic form, that is, a closed \(2\)-form on \(N\) which is nondegenerate in the sense that for any \(\rho \in N\), the map \(I_{\rho}\colon T_{\rho} N \to T^*_{\rho} N, I_{\rho}(s) = \sigma(s, \,\cdot\,)\) is bijective.

Example 1. The space \(\mathbb{R}^{2n}\) has a standard symplectic structure given by the \(2\)-form \(\sigma = dx_1 \wedge \,dx_{n+1} + \ldots + dx_n \wedge \,dx_{2n}\).

Example 2. More generally, if \(M\) is an \(n\)-dimensional \(C^{\infty}\) manifold, then \(N = T^* M\) becomes a symplectic manifold as follows: if \(\pi\colon T^* M \to M\) is the natural projection, there is a \(1\)-form \(\lambda\) on \(N\) (called the Liouville form) defined by \(\lambda_{\rho} = \pi^* \rho, \rho \in T^* M\). Then \(\sigma = d\lambda\) is a closed \(2\)-form. If \(x\) are local coordinates on \(M\), and if \((x,\xi)\) are associated local coordinates (called canonical coordinates) on \(T^* M\), then in these local coordinates \begin{align*} \lambda &= \xi_j \,dx^j, \\ \sigma &= d\xi_j \wedge \,dx^j. \end{align*} It follows that \(\sigma\) is nondegenerate and hence a symplectic form.

A Riemannian metric is an isomorphism $TM\to T^*M$, so it can be used to give a natural symplectic structure on the tangent bundle of a Riemannian manifold.

Definition. Let \((N,\sigma)\) be a symplectic manifold. Given any function \(f \in C^{\infty}(N)\), the Hamilton vector field of \(f\) is the vector field \(H_f\) on \(N\) defined by \[ H_f = I^{-1}(df) \] where \(df\) is the exterior derivative of \(f\) (a \(1\)-form on \(N\)), and \(I\) is the isomorphism \(TN \to T^* N\) given by the nondegenerate \(2\)-form \(\sigma\).

Example 2. In \( \mathbb{R}^{2n} \) one has \(I(s,t)=(t,-s)\), \(s, t \in \mathbb{R}^n\), and \[ H_f = \nabla_{\xi} f \cdot \nabla_x - \nabla_x f \cdot \nabla_{\xi}. \]

Definition. Let \((N,\sigma)\) be a symplectic manifold, and let \(f \in C^{\infty}(N)\). Denote by \(\varphi_t\) the flow on \(N\) induced by \(H_f\), that is, \[ \varphi_t\colon \rho(0) \mapsto \rho(t) \text{ where } \dot{\rho}(t) = H_f(\rho(t)). \]

Any Hamilton flow map is symplectic (\((\varphi_t)^* \sigma = \sigma\)) and consequently volume-preserving.

2. The geodesic flow

The geodesic flow on a Riemannian manifold $(M,g)$ is a dynamical system on $T^*M$ (or $TM$, the two are naturally isomorphic via the Riemann metric). A geodesic is uniquely determined by its initial position and velocity. The tangent bundle $T^*M$ is a symplectic manifold, and the geodesic flow can be realized as a Hamilton flow. It is given by the Hamilton function $$ f\colon T^* M \to \mathbb{R}, \ \ f(x,\xi) = \frac{1}{2} |\xi|_{g^{-1}}^2 = \frac{1}{2} g^{jk}(x) \xi_j \xi_k. $$ The Hamiltonian equation of motion becomes exactly the geodesic equation. One can also see the geodesic flow from the Langrangian point of view, or geometrically via local length minimization.

4. The Sasaki metric

The tangent bundle of a smooth manifold is a smooth manifold (of double dimension). There is a canonical Riemannian metric on the tangent bundle of a Riemannian manifold. This is the Sasaki metric.

The tangent bundle describes possible directions of motion on $M$; each point on $TM$ contains a point $x\in M$ and a vector $v\in T_xM$. Similarly, $TTM$ describes the directions of motion on $TM$. It is natural to split motion in two components: motion within a fiber (vertically) or motion of the base point only (horizontally). This division is most clear when $M=\R^n$; then $TM=\R^{2n}$ and $TTM=\R^{4n}$.

For any $\theta=(x,v)\in TM$, we split $T_\theta TM=H(\theta)\oplus V(\theta)$. Note that if $\dim(M)=n$, then $\dim(H(\theta))=\dim(V(\theta))=\dim(T_xM)=n$. It turns out that there are natural isomorphisms $H(\theta)\to T_xM$ and $V(\theta)\to T_xM$.

Let $\pi\colon TM\to M$ be the canonical projection. The vertical fiber is then $V(\theta)=\ker(d_\theta\pi)$ — there is no movement in the base.

There is a connection map $K\colon TTM\to TM$. A point $\theta\in TTM$ describes (to first order) a curve on $TM$. This is a curve on $M$ and a vector field along it. The covariant derivative of this vector field along this curve is $K(\theta)\in T_xM$. The horizontal fiber is then $H(\theta)=\ker(K_\theta)$ — there is no movement in the fiber (parallel transport).

The natural isomorphisms are $d_\theta\pi|_{H(\theta)}\colon H(\theta)\to T_xM$ and $K_\theta|_{V(\theta)}\colon V(\theta)\to T_xM$. The Sasaki metric is obtained by declaring these to be isometries (inherit the metric from $T_xM$) and $H(\theta)\perp V(\theta)$. We can split any vector $TTM\ni\eta=(\eta_h,\eta_v)$. In this notation $$ \ip{\eta}{\xi}_\Sa = \ip{\eta_h}{\xi_h} + \ip{\eta_v}{\xi_v}. $$

6. Coordinate representations of the Sasaki metric

For any coordinates on $M$, there are corresponding coordinates on $TM$ given by the coordinate functions and their differentials. If $x$ denotes the coordinates on $M$, let $(x,y)$ be the corresponding coordinates on $TM$. Let also $(x,y,X,Y)$ be the corresponding coordinates on $TTM$.

The vectors \(\{\delta_{x^j}=\partial_{x^j} - \Gamma^l_{jk} y^k \partial_{y^l}\}_{j=1}^n\) are a basis for the subspace \(H(\theta)\), where \(\theta =(x_0,y_0)\). The vectors \(\{\partial_{y^k}\}_{k=1}^n\) are a basis for the subspace \(V(\theta)\), \(\theta =(x_0,y_0)\).

The operators $d\pi$ and $K$ can be described in these coordinates: \begin{align*} K_\theta(\delta_{x^j}) &= 0,\\ d_\theta\pi(\delta_{x^j}) &= \partial_{x^j},\\ d_\theta\pi(\partial_{y^k}) &= 0,\\ K_\theta(\partial_{y^k}) &= \partial_{x^k}. \end{align*}

Given vectors \(\xi,\eta \in T_\theta TM\) and writing them in the basis given by $\delta_{x^j}$ and $\partial_{x^j}$, i.e. $$ \xi = X^i \delta_{x^i} + Y^k \partial_{y^k}, \quad \eta = \tilde X^i \delta_{x^i} + \tilde Y^k \partial_{y^k}, $$ we get that $$\langle \xi, \eta \rangle_{\text{Sasaki}} = \langle X^i \partial_{x^i}, \tilde X^i \partial_{x^i} \rangle + \langle Y^k \partial_{x^k}, \tilde Y^k \partial_{x^k} \rangle = g_{jk} X^j \tilde X^k + g_jk Y^j \tilde Y^k. $$