Questions tagged [isometry]
An isometry is a map between metric spaces that preserves the distance. This tag is for questions relating to isometries.
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isometry group of an orbifold is the quotient of the isometry group of its cover
I have a 2-dimensional orbifold that is the quotient of $\mathbb{R}^2$ under a group of isometries $\Gamma$ generated by $180^\circ$ rotations. I would like to say that $\text{Isom}(\Gamma \backslash ...
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Conditions to a linear map between tangent spaces being the differential of a isometry
I have the following situation: Suppose that we have two pseudo-Riemannian manifolds M and N, and fixed points $p$ $\in M$, $q \in N$, and a linear isometry between the corresponding tangent spaces ...
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Does the hyperbolic plane embed as a subgroup of its isometry group? As a submanifold?
Notation: let $\newcommand{\H}{\mathbb{H}^2} \H$ denote the hyperbolic plane, and let $\newcommand{\iso}{\operatorname{Iso}(\H)} \iso$ denote its isometry group. Fix an arbitrary$^\ast$ point $O \in \...
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Relation between isometries and curvature
Let $\gamma : I \to \mathbb{R}^3$ be a smooth regular curve, and let
$G_\gamma$ be the group of isometries that leave the image of $\gamma$ invariant.
It is well known that highly symmetric curves (...
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Isometry Group of a Finite-dimensional Normed Linear Space
I was studying the following discussion on Mathoverflow: Continuous automorphism groups of normed vector spaces, where it is mentioned that
"The isometry group of any (real) finite-dimensional ...
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Transition function of a hyperbolic atlas.
Let $\mathcal{S}$ be a hyperbolic surface without boundary and suppose $\Gamma$ is a subgroup of $Is^{+}(\mathcal{S})$ such that it acts properly discontinuously and is discrete. I wish to show that $\...
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Quotient of hyperbolic surface by a subgroup of orientation preserving isometries acting freely and properly discont. has a hyperbolic structure.
Take $\mathcal{S}$ to be a hyperbolic surface, $\Gamma < Is^{+}(\mathcal{S})$ such that it acts freely and properly discontinuously on $\mathcal{S}$. Then, $\mathcal{S}/\Gamma$ carries a uniquely ...
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Isometry group of the Klein Model
I want to show that the full Isometry group of the Klein Model $\mathbb{B}={[1:x:y]\in\mathbb{RP}^2}$, with the metric $d_{\mathbb{B}}(A,B)=\frac{1}{2}\log\frac{|BX|\,|AY|}{|AX|\,|BY|}$, is $\mathrm{...
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Rotating vectors to make their entries balanced
Let $v_1,\dots, v_d \in \mathbb{C}^d$ be a collection of vectors of length $1$ each. Does there always exist an isometry $U$ such that each of the vectors $Uv_i$ has entries (in a fixed basis) which ...
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Isometry group of $\mathbb{S}^2\times\mathbb{S}^2$ with a general Riemannian metric
Consider the Riemannian manifold $(\mathbb{S}^2\times\mathbb{S}^2,g)$, here $g$ is a general Riemannian metric, not necessarily a product metric. I want to know if there are some general results ...
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Help me by providing a few references on the study of isometric groups between two Banach spaces
I am interested in studying isometric groups between Banach spaces. Let $X$ and $Y$ be Banach spaces. Denote by $\operatorname{Iso}(X, Y)$ the collection of all surjective isometries from $X$ onto $Y$....
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Approximating Isometry locally and patching together
Consider two 2-dimensional Riemannian manifolds $(U,g)$ and $(\hat{U}, \delta)$ where $(x,y) \in U$. $\delta$ is the Euclidean metric and $$g = \begin{pmatrix}1 & 0\\0 & f(x)\end{pmatrix}.$$...
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Characterisation of symmetric spaces via dimension
Let $(M^n,g)$ be a simply-connected, complete Riemannian manifold. It is well known that the dimension of the isometry group of $(M,g)$ is bounded by $n(n+1)/2$, and this bound is attained if and only ...
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The order 4 isometry of the tetrahedron
Consider a tetrahedron centered at the origin. For example with the vertices $ (\pm1,0,-1/\sqrt{2}),(0,\pm1,1/\sqrt{2})$. The isometry group of the tetrahedron is $ S_4 $ and the subgroup of ...
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Why $T_{A(p)} S^n \subseteq T_p(S^n)$?
I have been trying to understand the solution to the following question:
Prove that the antipodal mapping $A: S^n \to S^n$ given by $A(p) = -p$ is an isometry of $S^n.$
And I found its solution online ...
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