# Questions tagged [isometries]

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### Isometries of fiber bundles

Let $F\to S\overset{\pi}{\to} B$ a Riemannian submersion with totally geodesic fibers. Question: How much information about the isometries of $S$ we have if we know the isometries of $F$ and $B$? For ...
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### Realizing mapping classes as isometries?

Let $\phi : M \to M$ be a diffeomorphism. Is there a metric $g$ on $M$ and a diffeomorphism $\psi$ isotopic to $\phi$ so that $\psi$ is an isometry with respect to $g$? I'm guessing the answer is no,...
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### Every partial isometry extends

I am interested in metric spaces $X$ where every isometry between two subsets of the space extends to a full isometry $X \to X$. Is there a name for this kind of space? Is there some paper which ...
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### Doubly-stochastic partial-isometric matrices

An $n\times n$ matrix $A$ with nonegative real entries $a_{ij}$ is said to be doubly stochastic if $\sum_{i=1}^na_{ij} = 1$, for all $j$, and $\sum_{j=1}^na_{ij}=1$, for all $i$. Much is known [1] ...
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### Can a knotted sphere isometrically embed into $\mathbb R^3$?

All smooth simple closed curves in $\mathbb R^3$ (knotted or not) can be isometrically embedded into $\mathbb R^2$ as a circle of equal arclength. The situation for knotted spheres seems more ...
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As is well-known, in dimension 2, a linear map $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a direct similarity if, once we identify $\mathbb{R}^2$ with $\mathbb{C}$, $f$ is of the form $$\forall z \... • 2,216 1 vote 1 answer 78 views ### Uniqueness of function with range \mathbb{S}^2 under a constraint Assume g,f\colon A\subset\mathbb{R}^M\rightarrow\mathbb{S}^2 are two bijective functions defined on the set A. Now assume a constraint C: \forall x,y\in A, \exists R\in SO(3)\colon Rf(x)=f(y)\... -1 votes 1 answer 109 views ### Isometric stratification preserves volume? Let K\subset \mathbb{R}^k be a non-empty compact subset let f:K \to K be Lipschitz and surjective. If, moreover, f is an isometry then clearly f preserves the Lebesgue measure of K. I ... • 4,775 2 votes 1 answer 516 views ### Separable Banach spaces isometric to quotient of a Banach space We know that every separable Banach space is isometrically isomorphic to a quotient space of (\ell^1,\|.\|_1). We also know that the norm defined by \|x\|=(\|x\|_1^2+\|x\|_2^2)^{1/2} for all x\in ... • 479 2 votes 1 answer 96 views ### Characterization of extrinsic distance prevserving embedding (see the definition given!) from low dimensional Euclidean spaces to high dimensions P.S. I asked the question on MSE more than a week ago, but didn't get any desired answer, so asking here. Let m < n \in \mathbb{N}. Let us equip \mathbb{R}^m, \mathbb{R}^n  with their ... • 1,417 1 vote 1 answer 128 views ### How does this orthogonality follow from the map being an isometry? This is a step of a proof in the book Variational Problems in Geometry by Seiki Nishikawa. I will ignore the background and change some of the statements and notations for simplicity. Let (M,g) ... • 273 -2 votes 1 answer 460 views ### Local isometry implies covering map: nonempty boundary case [closed] The following theorem is well known in the literature: Let M and N be riemannian manifolds and let f : M \to N be a local isometry. If M is complete and N is connected, then f is a ... • 2,085 14 votes 1 answer 848 views ### What are the applications of the Mazur-Ulam Theorem? Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, ... • 56.8k 2 votes 0 answers 143 views ### Spacetime symmetries We know some nice space-time have a lot of symmetries. It is said that Minkowski spacetime has$$ISO(d-1,1)/SO(d-1,1),$$de Sitter spacetime has$$SO(d,1)/SO(d-1,1)$$and anti-de Sitter spacetime ... • 2,137 4 votes 0 answers 77 views ### Proximal isometries in CAT(-1) metric space Let X be a rank 1 symmetric space of non-compact type and G its isometry group. G is a semisimple linear algebraic Lie group of non-compact type with trivial center. Let \rho be a ... • 445 3 votes 1 answer 407 views ### Pogorelov's rigidity theorem vs Cohn-Vossen rigidity theorem There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf Any isometry between two closed smooth convex surfaces (... • 21.2k 6 votes 1 answer 179 views ### Cohn-Vossen rigidity theorem in hyperbolic space There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf Any isometry between two closed smooth convex surfaces in ... • 21.2k 15 votes 1 answer 1k views ### Uniform distribution of points on Riemannian manifolds Recently, I came across a beautiful paper by Arnol'd and Krylov (Uniform distribution of points on a sphere...) that contains the following theorem: Theorem: Let A and B be two rotations of the ... • 1,054 1 vote 0 answers 65 views ### Mapping to distorted constant Gauss curvature surfaces of revolution There are three questions here. We imagine a flexible membrane that is scrolled out so as to straighten it. 1) How can we find a surface isometrically mapped from a surface of constant negative Gauss ... • 907 2 votes 0 answers 197 views ### Minkowski isometries Consider theorem 1.7 from chapter III of 'Elementary differential geometry' by O'Neill. It says that: Theorem 1.7: If \phi is an isometry of E^3 , then there exists a unique translation T and a ... 2 votes 0 answers 181 views ### The isometry groups of flag manifolds For any sequence of integers 0<n_1<...<n_k, there is a flag manifold of type (n_1, ..., n_k), which is the collection of ordered sets of vector subspaces of R^{(n_k)} (V_1, ..., V_k) ... • 10.4k 8 votes 2 answers 600 views ### If E\oplus_\phi E \cong E\oplus_\psi E, does it imply that \phi= \psi? Let E\neq \{0\} be a Banach space. For each p\in[1,\infty),  we define$$E\oplus_p E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \sqrt[p]{\|x\|^p + \|y\|^p}\}.$$Let F be another Banach space. By E\... • 603 2 votes 0 answers 41 views ### On the minimum distance along an orbit Let \Gamma be a nontrivial group of isometries of \mathbb{S}^n, n \geq 2, acting properly discontinuously. For p \in \mathbb{S}^n, define$$r(p) = \min_{g \in \Gamma \setminus\{e\} } d(p, g(p)...
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H$\vphantom{a}$i. Consider the Laplacian on $\mathbb R^n$, $$\Delta=\partial_i^2$$ It is easy to prove that the most general differential operator that commutes with rotations and translations is ...
This is a cross-post from MSE (no answer there). Let $M,N$ be oriented $d$-dimensional Riemannian manifolds, $M$ compact*, and let $f:M \to N$ be smooth. Consider the Dirichlet energy functional: \$...