Questions tagged [isometries]
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106 questions
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"Measuring" how far is one Banach space from being surjectively isometric to another
Bonjour/bonsoir à toutes et à tous.
Assume that $\mathbf{V} \equiv (V, \|\cdot\|_V)$ and $\mathbf{W} \equiv (W, \|\cdot\|_W)$ are Banach spaces (over the real or complex field).
Question 1. What ...
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Homometric $\Rightarrow$ isometric?
Suppose you know that there is a mapping between
two Riemmanian manifolds $M_1$ and $M_2$ such that,
for each $x_1 \in M_1$, the (codimension-1) measure of the set of points
at distance $d$ from $x_1$ ...
- 151k
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When do 0-preserving isometries have to be linear?
Let $\langle \mathbf{V},+,\cdot,||.|| \rangle$ be a normed vector space over $\mathbb{R}$.
Let $f : \mathbf{V} \to \mathbf{V}$ be an isometry that satisfies $f(\mathbf{0}) = \mathbf{0}$ .
What ...
user5810
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Must a surjective isometry on a dual space have a pre-adjoint?
Background: Let $X$ be a Banach space. We know a linear map $h$ is a surjective isometry of $X$ if and only if its adjoint $h^*$ is a surjective isometry of $X^*$.
In general, a linear map $g:X^* \...
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Terminology: "cocompact"
Let $M$ be a Riemannian manifold such that its isometry group $G=\textrm{Iso}(M)$ is a Lie group, and let $\Gamma$ be a subgroup of $G$.
1) What does the phrase "$\Gamma$ is a cocompact group of ...
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Testing for Riemannian isometry
In most physics situations one gets the metric as a positive definite symmetric matrix in some chosen local coordinate system.
Now if on the same space one has two such metrics given as matrices then ...
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