arXiv subjects
Publications and source records attributed to Matan Eilat.
We introduce two definitions with the purpose of quantifying the concept of a $C^{2,\alpha}$ surface for $0 < \alpha < 1$. The intrinsic definition is given in terms of the $\alpha$-H\"{o}lder norm of the Gauss curvature function. The extrinsic one relies on the existence of a smooth local representation of the Riemannian metric. We show that these definitions are equivalent up to a constant depending on $\alpha$.
Let $M$ be a $C^{2}$-smooth Riemannian surface. A classical theorem in differential geometry states that the Gauss curvature function $K : M \to \mathbb{R}$ vanishes everywhere if and only if the surface is locally isometric to the Euclidean plane. We give an asymptotically sharp quantitative version of this theorem with respect to an isothermal coordinate chart. Roughly speaking, we show that if $B$ is a Riemannian disc of radius $\delta > 0$ with $\delta^{2}\sup_{B}|K| < \varepsilon$ for some $0 < \varepsilon < 1$, then there is an isothermal coordinate map from $B$ onto an Euclidean disc of radius $\delta$ which is bi-Lipschitz with constant $\exp(4 \varepsilon)$.
Suppose that there exists a discrete subset $X$ of a complete, connected, $n$-dimensional Riemannian manifold $M$ such that the Riemannian distances between points of $X$ correspond to the Euclidean distances of a net in $\mathbb{R}^{n}$. What can then be derived about the geometry of $M$? In arXiv:2004.08621 it was shown that if $n=2$ then $M$ is isometric to $\mathbb{R}^{2}$. In this paper we show two consequential geometric properties that the manifold $M$ shares with the Euclidean space in any dimension. The first property is that $X$ is a net with respect to the Riemannian distance in $M$. The second property is that all geodesics in $M$ are distance minimizing, and there are no conjugate points in $M$. This demonstrates the possibility of inferring infinitesimal qualities from discrete data, even in higher dimensions. As a corollary we obtain that the large-scale geometry of $M$ is asymptotically Euclidean.
Let $M$ be a complete, connected Riemannian surface and suppose that $\mathcal{S} \subset M$ is a discrete subset. What can we learn about $M$ from the knowledge of all distances in the surface between pairs of points of $\mathcal{S}$? We prove that if the distances in $\mathcal{S}$ correspond to the distances in a $2$-dimensional lattice, or more generally in an arbitrary net in $\mathbb{R}^2$, then $M$ is isometric to the Euclidean plane. We thus find that Riemannian embeddings of certain discrete metric spaces are rather rigid. A corollary is that a subset of $\mathbb{Z}^3$ that strictly contains $\mathbb{Z}^2 \times \{ 0 \}$ cannot be isometrically embedded in any complete Riemannian surface.