Let $M$ be a compact, connected, oriented smooth Riemannian manifold with non-empty boudary. Let $f:M \to M$ be a smooth orientation preserving isometric immersion. Is it true that $f$ is locally distance preserving? (in the sense that around every $p \in M$ there exist a neighbourhood $U$ s.t $d(f(x),f(y))=d(x,y)$ for all $x,y \in U$) This […]

Let $(X,d)$ be a metric space. There are always other metrics on $X$ which generates the same topology, and have the same isometry groups, for instance $\tilde d =\sqrt d$. (The same will be true for any injective continuous function $f:\mathbb{R}^{\ge 0}\to \mathbb{R}^{\ge 0}$, taking zero to itself and which satisfying some constraints to ensure […]

This is probably very elementary, yet I am not sure how to approach this. Suppose $f$ is a conformal map between Riemannian manifolds, which maps geodesics to geodesics. (i.e if $\alpha$ is a geodesic, then $f \circ \alpha$ is a geodesic). Is it true that $f$ must be a “scaled isometry“? (Does there exist a […]

Let $M,N$ be smooth compact oriented Riemannian manifolds with boundary. Suppose that both $M,N$ can be isometrically immersed in each other. Must $M,N$ be isometric? Does anything change if we also assume $\operatorname{Vol}(M)=\operatorname{Vol}(N)$? Note: I assume $M,N$ are connected (Otherwise, as mentioned by Del, we can take $N$ to be two disjoint copies of $M$). […]

Let $X$ be a metric space with at least $5$ points such that any five point subset of $X$ can be isometrically embedded in $\mathbb R^2$ , then is it true that $X$ can also be isometrically embedded in $\mathbb R^2$ ?

I am doing Isometries i.e. Rotations, translations and reflections at the moment. General equations for 2 of them are “easy” Translation: $f(x,y)=(x+a,y+b)$ Rotation about origin through the angle $\alpha$ is just $f(r,\theta)=(r,\theta+\alpha)$ However I struggle to find similiar function for a reflection in a line. For some simple cases like Reflection in line $y=x$ this […]

Let $G$ be the group of all isometries of $C$ consisting of all real translations, and all glide reflections with axis $\mathbb R$. Show that $G$ is isomorphic to $\{1,-1\} \times \mathbb R$. I feel it difficult, first of all, to write down the group $G$. I think all real translations can be represented as […]

I am wondering the following question: Let $X$ be a separable Banach space which is linearly isomorphic to a dual Banach space $Y^*$. Is there a Banach space $Z$ such that $X$ is lineraly isometric to the dual of $Z$: $X=Z^*$. I think that the answer is no, but I do not have a counterexample. […]

$\newcommand{\<}{\langle} \newcommand{\>}{\rangle} $Let $||\cdot||$ be a norm on a finite dimensional real vector space $V$. Does there always exist some inner product $\<,\>$ on $V$ such that $\text{ISO}(|| \cdot ||)\subseteq \text{ISO}(\<,\>)$ ? Update: As pointed by Qiaochu Yuan the answer is positive. This raises the question of uniqueness of the inner product $\<,\>$ which satisfies […]

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. (You can see this question, where it is proved that $T$ preserves some inner product on $V$ if and only if $V$ admits a […]

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