Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 14 / 3

A simple Riemannian manifold has strictly convex boundary and a unique geodesic between each pair of points, smoothly dependent on its endpoints; there are no conjugate points and no trapped geodesics. Let $\nu$ denote the outward unit normal. For a boundary unit vector with $⟨v,\nu ⟩<0$, follow its unit-speed geodesic ${\gamma }_{x,v}$ until the next boundary intersection at time $\tau (x,v)>0$. Its geodesic scattering relation isThus it maps incoming boundary unit vectors to outgoing ones, recording both exit point and direction.

If the scattering relation is defined on the full boundary unit bundle, extend it on outgoing vectors by following the same oriented geodesic backwards to its previous boundary intersection, and fix tangential vectors. This extension is an involution, because the second application follows the same segment back to its starting point with the same velocity. The exit time is not a component of the scattering relation itself; together with it one obtains lens data. For a simple Riemannian metric, $\tau$ is the boundary distance between the endpoints of the segment.

First the common boundary distance function determines the induced tangential Riemannian metric. For a boundary curve $c(t)$ with $c(0)=x$ and $c^{\prime }(0)=w\in T_x\partial M$, the local first-order distance expansion givesFor clarity, the upper bound follows from the length of the boundary curve. In a small coordinate neighbourhood the Riemannian metric is arbitrarily close to its value at $x$, so the length of any sufficiently short joining path is bounded below by that constant-metric displacement norm with an arbitrarily small relative error. A minimizing short path cannot leave that neighbourhood. These two bounds prove the limit. Equality of the boundary distance functions then gives $|w{|}_{g_1}=|w{|}_{g_2}$. The polarization identity yieldsThis alone fixes no choice of normal coordinates, so a boundary-fixing gauge is needed to compare the full tangent-space Riemannian metrics.

Let $n_i$ be the inward unit normals and letbe sufficiently small boundary normal coordinates. Their pullbacks have the form $C_i^{\ast }{g}_i=d{r}^2+h_i(r)$, with $h_i(0)=h$. On a collar define $\psi =C_2\circ C_1^{-1}$. It fixes the boundary, has $D\psi (w)=w$ for tangential $w$, and sends $n_1$ to $n_2$. For tangent-space vectors $w+a{n}_1$ and $z+b{n}_1$ at the boundary, therefore,This proves the desired full Riemannian metric equality locally at the boundary.

We also justify extending this local identification to a global diffeomorphism. Interpolate the positive Riemannian metrics by $g_s=(1-s)g_1+s{g}_2$, for $0\le s\le 1$. Let $C_s$ be their inward unit-normal exponential collars. Compactness of the boundary and parameter interval gives a common small collar width for all $s$. The intermediate Riemannian metrics need not be simple; only these local collars are used.

On the image of $C_s$, define the time-dependent vector fieldwhere $\chi$ is one near zero and vanishes before the outer edge of the collar; extend $Y_s$ by zero elsewhere. This is a smooth vector field on $M$, vanishing at the boundary because $C_s(x,0)=x$. Its time-dependent flow ${\Psi }_s$ is a global diffeomorphism fixing the boundary. For every sufficiently small fixed $r$, the curve $s\mapsto C_s(x,r)$ solves its equation, since $\chi (r)=1$. Uniqueness gives ${\Psi }_s(C_0(x,r))=C_s(x,r)$. Consequently ${\Psi }_1$ agrees near the boundary with $C_2{C}_1^{-1}$. Taking $\psi ={\Psi }_1$ proves

Full equality of the Riemannian metrics at the boundary identifies their boundary unit bundles, outward normals and induced boundary Riemannian metrics. We show directly that the common boundary distance function determines both endpoint directions.

For distinct boundary points $x,y$, let $\gamma$ be the unique joining unit-speed geodesic, with initial velocity $v$ and final velocity $w$. The first variation of its length gives, for $a\in T_x\partial M$ and $b\in T_y\partial M$,Indeed, integration by parts in the length variation leaves just the endpoint terms, because the interior term contains ${\nabla }_{\dot{\gamma }}\dot{\gamma }=0$. The Riemannian metric is simple, so the off-diagonal endpoint dependence is smooth and these derivatives are valid.

Thus the tangential components are $v_T=-{\mathrm{grad}}_x^{\partial M}{d}_g(x,y)$ and $w_T={\mathrm{grad}}_y^{\partial M}{d}_g(x,y)$. Unit speed and the incoming/outgoing signs supply the missing normal components:All quantities on the right use only boundary distance and the common boundary Riemannian metric. Every strictly incoming vector for $g_1$ has some exit point $y$; the geodesic joining the same pair for $g_2$ has exactly the same reconstructed initial and final vectors. Hence both geodesic scattering relations give the same exit for that input. Tangential vectors are fixed, and the outgoing extensions agree as inverses. ThereforeIn fact their exit times also agree, since each is $d_g(x,y)$.