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Let $M$ be a compact, connected, oriented, smooth Riemannian manifold with non-empty boundary. Let $f:M \to M$ be a smooth orientation preserving isometric immersion. Is it true that $f(\partial M) \ …

This is a cross-post. While working on a variational problem, I have reached to the following question. Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=1$, and let $D \subseteq \mathbb{R}^2$ be t …

$\newcommand{\SO}[1]{\text{SO}(#1)}$ $\newcommand{\Hom}[1]{\text{Hom}(#1)}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\S}{\mathbb{S}}$ The question in a nutshell: Are the "best" spherical maps local …

Are there any smooth manifolds $M$ with the following property: There exist a realizing metric $d$ (i.e $d$ induces the topology on $M$), and $d$ is smooth on all of $M \times M$? If not, is it po …

I have posted this question in mathSE a few weeks ago (and proposed a bounty) but so far got no response. In the book Riemanian geometry (by do-Carmo), the following result is proved (Corollary 2.9 p …

I am completenig some details based on Anton's answer: We prove the following theorem: Let $X,Y$ be Riemannian manifolds, and let $f:X \to Y$ be a length preserving map. Then $df$ is an isometry alm …

$\newcommand{\al}{\alpha}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\e}{\epsilon}$ Let $X,Y$ be Riemannian manifolds, such that $\dim(X) > \dim(Y)$. I am trying to prove the following statement (ment …

Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth). Suppose we have a map $\phi:M \to N$ which satisfies the following properties: $$(1) \, \ …

(This question is a special case of a question I asked at SE, which got no answer there) Let $M,N$ be diffeomorphic connected compact Riemannian manifolds, and let $f:M \to N$ be a surjective nonexp …

This paper might give you some ideas on how to calculate the geodesics. It is about left invariant metrics on $GL_n(\mathbb{R})$. The geodesics are calculated using their characterization as critical …

$\newcommand{\al}{\alpha}$ The answer is no, there are many more such metric which are not induced by a Riemannian metric. (This answer is based on the comments above, made by user89334). Examples: …

Consider the group $GL_n(\mathbb{R})$ with its standard topology. It is not hard to show that there exists Riemannian metrics on it which are left-$GL_n$ and right-$O_n$ invariant. (In fact it's pos …

$\newcommand{\til}{\tilde}$ Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds. Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open Riem …