Related papers: Simple Riemannian surfaces are scattering rigid

We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the…

We consider a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. This is called an $\mathcal{MP}$-system. On simple $\mathcal{MP}$-systems, we consider both the boundary rigidity problem and the…

We study the scattering rigidity problem for standard stationary manifolds using timelike geodesics with a fixed momentum. Taking advantage of the symmetry of this manifolds, we use Hamiltonian reduction to show that this problem is related…

We study the boundary rigidity problem for compact Riemannian manifolds with boundary $(M,g)$: is the Riemannian metric $g$ uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function $\rho_g(x,y)$…

We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci…

We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of $2 N$ dimensional Ricci flat…

We prove the following rigidity theorem: For an n-dimensional compact Riemannian manifold with boundary whose Ricci curvature is bounded by n-1 from below, if its boundary is isometric to the standard sphere of dimension n-1 and totally…

In this work, we prove the following three rigidity results: (i) in a real-analytic globally hyperbolic spacetime $(M,g)$ without boundary, the time separation function restricted to a thin exterior layer of a unknown compact subset $K…

It was proved in \cite{NS1} that obstacles $K$ in $\R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also…

The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove a theorem claimed by Lusternik: in any Riemannian 2-sphere whose simple length spectrum consists of only one element L, any…

In this paper we analyze the local and global boundary rigidity problem for general Riemannian manifolds with boundary $(M,g)$. We show that the boundary distance function, i.e., $d_g|_{\partial M\times\partial M}$, known near a point $p\in…

We observe that the maximal open set of constant curvature k in a Riemannian manifold with curvature bounded below or above by k has a convexity type property, which we call "two-convexity". This statement is used to prove a number of…

We prove that if two non-trapping obstacles in $\mathbb{R}^n$ satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length…

We show that round hemispheres are the only compact 2 dimensional Riemannian manifolds (with or without boundary) such that almost every pair of complete geodesics intersect once and only once. We prove this by establishing a sharp…

This paper constructs a Riemann surface associated to the icosahedron and discusses the geodesics associated to a flat metric on this surface. Because of the icosahedral symmetry, this is a distinguished special case of the example treated…

The question whether a Riemannian manifold is geodesically connected can be studied from geometrical as well as variational methods, and accurate results can be obtained by using the associated distance and related properties of the…

We study an inverse problem for a non-compact Riemannian manifold whose ends have the following properties : On each end, the Riemannian metric is assumed to be a short-range perturbation of the metric of the form $(dy)^2 + h(x,dx)$,…

We state that any constant curvature Riemannian metric with conical singularities of constant sign curvature on a compact (orientable) surface $S$ can be realized as a convex polyhedron in a Riemannian or Lorentzian) space-form. Moreover…

For a compact Riemannian manifold with boundary, endowed with a magnetic potential $\alpha$, we consider the problem of restoring the metric $g$ and the magnetic potential $\alpha$ from the values of the Ma\~n\'e action potential between…

In this paper, we show that simple, thick negatively curved two-dimensional P-manifolds, a large class of surface amalgams, are marked length spectrum rigid. That is, if two piecewise negatively curved Riemannian metrics (satisfying certain…