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$\newcommand{\id}{\text{id}}$ $\newcommand{\Hom}{\text{Hom}}$ This is a cross-post. Let $V$ be a $d$-dimensional real vector space, and let $2 \le k \le d-1$. Every inner product on $V$ induces an inn …

This question is a special case of this one. Let $s(\theta)>0, b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$. Do there exist a diffeomorphism $\phi:\mathbb{S}^1 \to \ma …

Let $V$ be a finite dimensional vector space. Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry. We know $T …

I have decided to write the details of the solution suggested by Sawin, for sake of completeness. Let $M=\mathbb{S}^1\times\mathbb{R}$. (i.e $M$ is the infinite cylinder in $\mathbb{R}^3$). $\phi:M …

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 …

Let $f \neq Id$ be a diffeomorphism (of a smooth manifold $M$) which admits some Riemannain metric on $M$ making it an isometry. How many different metrics are preserved by $f$? Note that $Met(f)=\{g …

I am just adding a few details to Vladimir's answer: Lemma: Assume there exists a point $x$ such that the orbit w.r.t. the iterations of the $\phi$ (i.e., the set $\{x,ϕ(x),ϕ(ϕ(x)),...\}$ is dense in …

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 …

Let $M$ be a smooth manifold with non-empty boundary. Let $g$ be a smooth Riemannian metric on $M$. Is the following true? For every $\epsilon >0$ there exist a Riemannian metric $g_{\epsilon}$ w …

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M, \N$ be smooth two-dimensional Riemannian manifolds. Are there any local obstructions for the existence of a smooth map $f:\M \to …

$\newcommand{\M}{M}$ $\newcommand{\N}{N}$ $\newcommand{\TM}{TM}$ $\newcommand{\TN}{TN}$ $\newcommand{\TstarM}{T^*M}$ $\newcommand{\Ga}{\Gamma}$ Let $\M,\N$ be smooth manifolds, $\phi:\M \to \N$ be a …

$\newcommand{\id}{\operatorname{Id}}$ Well, there is a natural way to view this "pullback-symmetry": Exterior derivative commutes with pullbacks: Let $f:M \to N$ be a smooth map, $E$ a vector bundl …

Let $h:\mathbb{S}^2 \to \mathbb{R} $ be a smooth function satisfying: $h(-x)=-h(x)$ For every hemisphere $A \subseteq \mathbb{S}^2$, $\int_{A}h\text{Vol}_{\mathbb{S}^2}=0$, where $\text{Vol}_\mathbb …

$\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\N}{\mathc …

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be $d$-dimensional oriented Riemannian manifolds ($d \ge 2$). Let $f:\M \to \N$ be smooth. Let $1 \le k \le d-1$ be fixe …