12.09.2018

# Geodesic flows: notes 7

## Geodesic flows, notes 7: recap


## 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 $$\langle \xi, \eta \rangle_{\text{Sasaki}} = \langle X^i \partial_{x^i}, \tilde X^i \partial_{x^i} \rangle + \langle Y^k \partial_{x^k}, \tilde Y^k \partial_{x^k} \rangle = g_{jk} X^j \tilde X^k + g_jk Y^j \tilde Y^k.$$