Lens rigidity for manifolds with hyperbolic trapped set

By C. Guillarmou. 2014

81 citations recorded by the source corpus.

For a Riemannian manifold $(M,g)$ with strictly convex boundary $\partial M$, the lens data consists in the set of lengths of geodesics $\gamma$ with endpoints on $\partial M$, together with their endpoints $(x_-,x_+)\in \partial M\times \partial M$ and tangent exit vectors $(v_-,v_+)\in T_{x_-} M\times T_{x_+} M$. We show deformation lens rigidity for a large class of manifolds which includes all manifolds with negative curvature and strictly convex boundary, possibly with non-trivial topology and trapped geodesics. For the same class of manifolds in dimension $2$, we prove that the set of endpoints and exit vectors of geodesics (ie. the scattering data) determines the topology and the conformal class of the surface.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9e508fe00462b02fd0e466d1f1a76fd95c35c7e9

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Citations in the corpus

Invariant distributions and X-ray transform for Anosov flows (2014, unknown): “To motivate even more this new way of analyzing X-ray transform, we mention that using tools of similar nature, we are able to prove in [Gu] the injectivity of X-ray transform for all negatively curved manifold with strictly convex boundary, and that the scattering map for the flow on negatively cur”
Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge (2017, background): “In a recent work, Guillarmou [8] proved that the lens data determine the conformal class for Riemannian surfaces with hyperbolic trapped sets, no conjugate points and strictly convex boundary, and deformational rigidity in all dimensions under these conditions.”
Dynamical zeta functions for Anosov flows via microlocal analysis (2013, methodology): “velopment of new microlocal methods as the escape on the cotangent bundle can occur both at ber innity (as in this paper) and at the manifold innity. A surprising application was given by Guillamou [G2] who established deformation lens rigidity for a class of DYNAMICAL ZETA FUNCTIONS VIA MICROLOCAL ”