A Riemannian metric on a manifold allows to measure the length of tangent vectors, and the angles between pairs of tangent vectors. From there, it further allows to measure the length of curves, and more generally the volume of compact submanifolds. The subject of "geometric inverse problems" in Riemannian geometry addresses several specific versions of the following questions:
- In a compact Riemannian manifold with non-empty boundary, what can we infer on the inner geometry of the manifold from measurements that can be made at the boundary?
- Analogously, in a closed Riemannian manifold, what can we infer on the geometry of the manifold from so-called spectral measurements?
In this colloquium talk, I will make these questions more precise, and present a few answers, ranging from classical results to more modern ones. I will focus on measurements involving geodesics, that is, curves that are locally length minimizing.
The geodesics of a Riemannian manifold are described by a dynamical system: the geodesic flow. This is a special instance of a more general class of dynamical systems on contact manifolds, called Reeb flows. Time permitting, I will also address analogous geometric inverse problems in this more general class.
Zoom link: https://univ-cotedazur.zoom.us/j/82130421344?pwd=N1RGWWIrY2VGb3hMY0RqcGJBZCtqZz0
