Determination of a compact Finsler manifold from its boundary distance map and an inverse problem in elasticity

Abstract

We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differentiable structures. We construct the optimal fiberwise open subset of its tangent bundle and show that the boundary distance map determines the Finsler function in this set but not in its exterior. If the Finsler function is fiberwise real analytic, it is determined uniquely. We also discuss the smoothness of the distance function between interior and boundary points. We recall how the fastest $qP$-polarized waves in anisotropic elastic medium are a given as solutions of the second order hyperbolic pseudodifferential equation $(\frac{\partial ^{2}}{\partial t^{2}}-\lambda ^{1}(x,D))u(t,x)=h(t,x)$ on ${\mathbb R}^{1+3}$, where $\sqrt{\lambda ^{1}}$ is the Legendre transform of a fiberwise real analytic Finsler function $F$ on ${\mathbb R}^{3}$. If $M \subset {\mathbb R}^{3}$ is a $F$-convex smooth bounded domain we say that a travel time of $u$ to $z \in \partial M$ is the first time $t>0$ when the wavefront set of $u$ arrives in $(t,z)$. The aforementioned geometric result can then be utilized to determine the isometry class of $(\overline M,F)$ if we have measured a large amount of travel times of $qP$-polarized waves, issued from a dense set of unknown interior point sources on $M$.