DIFFERENTIAL GEOMETRY/PDE SEMINAR

Wednesday, January 31 Padelford C-36 3:45 PM

On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points

Vladimir Sharafutdinov (Sobolev Institute of Mathematics)

Let (M, g) be a compact Riemannian manifold with boundary. The boundary rigidity problem consists of determining the Riemannian metric g by measuring the geodesic distance, d_{g}, between boundary points. (M, g) is called boundary rigid if the metric g is uniquely determined by d_{g} up to isometries which are the identity on the boundary.

We consider the linearized problem. A smooth one-parameter family g^{τ} (−ε < τ < ε) of metrics on a manifold M is said to be a deformation of a metric g if g^{0} = g. Such a family is called a boundary-distance preserving deformation if the boundary distance function d_{g^{τ}} of the metric g^{τ} is independent of τ. A deformation g^{τ} is called the trivial deformation if there exists a family φ^{τ} of diffeomorphisms of

M wich are the identity on the boundary and such that g^{τ} = (φ^{τ})^{∗}g. A manifold (M, g) is called deformation boundary rigid if it does not admit a nontrivial boundary-distance preserving deformation.

Theorem 1 A compact simply simply connected two-dimensional Riemannian manifold with strictly convex boundary and with no focal points is deformation boundary rigid.

We consider also the corresponding question for closed (= compact without boundary) Riemannian manifolds. Given such a manifold (M, g), a deformation g^{τ} of the metric g is called an isospectral deformation if the spectrum of the Laplace — Beltrami operator Δ^{τ} of the metric g^{τ} is independent of τ. A closed manifold (M, g) is called spectrally rigid if it does not admit a nontrivial isospectral deformation.

Theorem 2 A closed two-dimensional Riemannian manifold with geodesic flow of Anosov type and with no focal points is spectrally rigid.

In the first part of our talk we will discuss the relationship between boundary rigidity, deformation boundary rigidity, and spectral rigidity. Then the above-stated theorems will be reduced to some inverse problem for the kinetic equation. The latter problem will be treated in the second part of our talk which is to be held on Monday at the Inverse Problem Seminar.