Related papers: Reshetnyak rigidity for Riemannian manifolds

The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds of arbitrary index, using one-sided bounds on the Riemann tensor which in the Riemannian…

One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincar\'e…

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…

Although the Nash theorem solves the isometric embedding problem, matters are inherently more involved if one is further seeking an embedding that is well-behaved from the standpoint of submanifold geometry. More generally, consider a…

In Gromov's treatise Partial Differential Relations (volume 9 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 1986), a continuous map between Riemannian manifolds is called isometric if it preserves the length of rectifiable…

Inspired by a recent work of Wang-Zhao, in this note we prove that for a fixed $n$-dimensional closed Riemannian manifold $(M^n, g)$, if an $\mathrm{RCD}(K, n)$ space $(X, \mathsf{d}, \mathfrak{m})$ is Gromov-Hausdorff close to $M^n$, then…

We study metric spaces homeomorphic to a closed oriented manifold from both geometric and analytic perspectives. We show that such spaces (which are sometimes called metric manifolds) admit a non-trivial integral current without boundary,…

We show, as our main theorem, that if a Lipschitz map from a compact Riemannian manifold $M$ to a connected compact Riemannian manifold $N$, where $\dim M \geq \dim N$, has no singular points on $M$ in the sense of F.H. Clarke, then the map…

We prove rigidity results describing contextually-constrained maps defined on Grassmannians and manifolds of ordered independent line tuples in finite-dimensional vector or Hilbert spaces. One statement in the spirit of the Fundamental…

Let $\Sigma$ be a Riemannian manifold with strictly convex spherical boundary. Assuming absence of conjugate points and that the trapped set is hyperbolic, we show that $\Sigma$ can be isometrically embedded into a closed Riemannian…

In this article we introduce conformal Riemannian morphisms. The idea of conformal Riemannian morphism generalizes the notions of an isometric immersion, a Riemannian submersion, an isometry, a Riemannian map and a conformal Riemannian map.…

We show that under certain conditions, a nontrivial Riemannian submersion from positively curved four manifolds does not exist. This gives a partial answer to a conjecture due to Fred Wilhelm. We also prove a rigidity theorem for Riemannian…

We study the rigidity of compact submanifolds of Riemannian manifolds of arbitrary codimension that satisfy a sharp pinching condition involving the norm of the second fundamental form and the mean curvature. Without assuming that the…

These notes provide an exposition on obtaining the well-known standard results of quasiregular maps on Riemannian manifolds, given the corresponding theory in the Euclidean setting. We recall several different approaches to first-order…

We establish a global rigidity theorem for Riemannian metrics without conjugate points on three-manifolds of the form $M = \Sigma \times S^1$, where $\Sigma$ is a compact orientable surface of genus at least 2. The main result states that…

We establish a localized Bochner-type rigidity theorem for harmonic maps between Riemannian manifolds. Let $f : (M,g) \to (\overline{M},\overline{g})$ be a harmonic map from a compact manifold. Instead of assuming a global nonpositivity…

In this paper, we address several interconnected problems in the theory of harmonic maps between Riemannian manifolds. First, we present necessary background and establish one of the main results of the paper: a criterion characterizing…

We determine when a quasi-isometry between discrete spaces is at bounded distance from a bilipschitz map. From this we prove a geometric version of the Von Neumann conjecture on amenability. We also get some examples in geometric groups…

We prove that every 1-Lipschitz map from a closed metric surface onto a closed Riemannian surface that has the same area is an isometry. If we replace the target space with a non-smooth surface, then the statement is not true and we study…

In this paper, we obtain the following generalisation of isometric $C^1$-immersion theorem of Nash and Kuiper. Let $M$ be a smooth manifold of dimension $m$ and $H$ a rank $k$ subbundle of the tangent bundle $TM$ with a Riemannian metric…