arXiv research

Study scattering rigidity on stationary manifolds using geodesics.

Paper proves rigidity theorems for geodesically reversible Finsler metrics.

Totally geodesic subvarieties in moduli space are locally rigid.

The paper proves rigidity for Anosov geodesic flows on certain manifolds.

Two Riemannian manifolds are said to have $C^k$-conjugate geodesic flows if there exist an $C^k$ diffeomorphism between their unit tangent bundles which intertwines the geodesic flows. We obtain a number of rigidity results for the geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step nilmanifold…

Study geodesic curvature in 2D Alexandrov spaces, generalizing results from spaces with curvature above.

Study inverse problems for twisted geodesic flows on manifolds.

Study rigidity of geodesic balls on manifolds with boundary.

Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.

We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric $g$ is geodesically rigid: if any other complete metric $\bar g$ has the same (unparametrized) geodesics with $g$, then the Levi-Civita connections of $g$ and $\bar g$ coincide.

Boundary rigidity defined for hyperbolic spaces, tied to geometric properties.

We prove some rigidity results on geodesic orbit Finsler spaces with non-positive curvature. In particular, we show that a geodesic Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.

Researchers prove rigidity of 2D manifolds from boundary geodesic lengths.

Proves rigidity of geodesic balls in spheres under certain deformations.

We introduce the notion of a topological geodesic in a 3-manifold. Under suitable hypotheses on the fundamental group, for instance word-hyperbolicity, topological geodesics are shown to have the useful properties of, and play the same role in several applications as, geodesics in negatively curved spaces. This permits…

New metric on geodesic currents connects different surface genera.

Researchers refine local rigidity for marked length spectrum and introduce a new pressure metric.

Scattering rigidity of a Riemannian manifold allows one to tell the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to tell the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results…

Floating geodesic planes in Hitchin manifolds have fractal closures with non-integer dimensions.

New theorem shows certain curved surfaces are uniquely identified by their geodesic lengths.

In this paper, we prove a scalar curvature rigidity result for geodesic balls in S^n. This result contrasts sharply with the recent counterexamples to Min-Oo's conjecture for the hemisphere (cf. [5]).

Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n>3 then the holonomy representation of pi_1 (W) into the isometry group of hyperbolic n-space is infinitesimally rigid.

Study rigidity in Penrose's singularity theorem with weakly trapped surfaces.

New distance function proves rigidity in geodesic lamination space.

Study shows X-ray transform injectivity for hyperbolic manifolds.

Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n>3. This paper verifies this conjecture for a specific example based on the 4-dimensional hyperbolic 120-cell.

New counterexample shows curved surfaces can deform geodesics without diffeomorphism.

Paper proves rigidity of spherical ring patterns on surfaces.

Study of centralizer elements preserving geodesic flow foliations on covers.

New characterizations for manifolds with boundary rigidity results.

The paper proves unique determination of Dehn fillings in hyperbolic 3-manifolds.

Totally geodesic hypersurfaces in hyperbolic manifolds are rigid under certain conditions.

Study magnetic geodesics on Heisenberg groups and manifolds.

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we…

Entropy rigidity shows volume bounds for convex projective manifolds.

This paper classifies geodesics of projectively flat sprays and introduces a method to determine sprays based on geodesics.

Study rigidity of geodesic planes in geometrically finite manifolds with cusps.

We consider the scattering and lens rigidity of compact surfaces with boundary that have a trapped geodesic. In particular we show that the flat cylinder and the flat Möbius strip are determined by their lens data. We also see by example that the flat Möbius strip is not determined by it's scattering data. We then cons…

Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.

The paper proves uniqueness of closed timelike geodesics on de-Sitter tori with one singularity.

Paper investigates rigidity of synthetic Ricci curvature bounds on manifolds.

New rigidity theorem for manifolds with specific negative curvature.

The paper proves a rigidity theorem for null geodesic travel times in asymptotically AdS spacetimes.

The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.

Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthog…

In this paper we consider the lens rigidity problem with partial data for conformal metrics in the presence of a magnetic field on a compact manifold of dimension $\geq 3$ with boundary. We show that one can uniquely determine the conformal factor and the magnetic field near a strictly convex (with respect to the magne…

Geometric study of Outer Space using envelopes and geodesics.

In this work we show that for the geodesic spray $S$ of a Finsler function $F$ the most natural projective deformation $\widetilde{S}=S -2 λF\mathbb C$ leads to a non-Finsler metrizable spray, for almost every value of $λ\in \mathbb R$. This result shows how rigid is the metrizablility property with respect to certain …