Scattering rigidity for analytic metrics

Abstract

For analytic negatively curved Riemannian manifold with analytic strictly convex boundary, we show that the scattering map for the geodesic flow determines the manifold up to isometry. In particular one recovers both the topology and the metric. More generally our result holds in the analytic category under the no conjugate point and hyperbolic trapped sets assumptions.

Local lens rigidity for manifolds of Anosov type

“…The fact that I g 0 2 is injective on divergence-free tensors was proved in [Gui17b] in non-positive curvature and in general on Anosov surfaces by [Lef19b] (without any assumption on the curvature). It was also proved in [GGJ22] that I g 0 2 is injective for real-analytic metrics g 0 which implies that generic smooth metrics of Anosov type have an injective X-ray transform operator I g 0 2 ; generic injectivity of I g 0 2 follows from the work of the first and third authors [CL21] as well, admitting also Theorem 1.10 below. As a corollary of Theorem 1.8, we obtain: Corollary 1.9.…”

“…(3) In dimension n ≥ 2, on all real analytic manifold of Anosov type, injectivity of I g 2 is proved in [GGJ22]. We conjecture that the following holds: Conjecture 3.4 (Solenoidal injectivity of the X-ray transform on manifolds of Anosov type).…”

“…(3) In dimension n ≥ 3, Stefanov-Uhlmann-Vasy [SUV21] prove that for general metrics with strictly convex boundary, the lens data determines the metric in a neighborhood of ∂M ; applying this result in the setting of negatively curved manifold, one can recover the metric outside the convex core of the manifold (which contains the projection of the trapped set). (4) In [GGJ22], Bonthonneau, Jézéquel, and the second author recently proved Conjecture 1.7 under the extra assumption that (M 1 , g 1 ), (M 2 , g 2 ) are real analytic, but only using the equality S g 1 = S g 2 of the scattering maps. Our first result in this article is the following local rigidity result answering Conjecture 1.7 for metrics close to each other.…”