All Questions
Tagged with isometries riemannian-geometry
10 questions
16
votes
2
answers
967
views
Tweetable way to see Riemannian isometries are harmonic?
$\newcommand{\al}{\alpha}$
$\newcommand{\euc}{\mathcal{e}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\Det}{\operatorname{Det}}$
Smooth Riemannian isometries are harmonic. Can one conclude ...
- 6,741
12
votes
1
answer
2k
views
Isometry group of pseudo Riemannian manifold always a Lie group? (Myers-Steenrod)
Myers-Steenrod states that the isometry group of a Riemannian manifold is a Lie group. Is that also true for pseudo Riemannian manifolds? I didn't find anything related to that.
Cheers
- 123
25
votes
6
answers
3k
views
Isometric embedding of SO(3) into an euclidean space
Consider $SO(3)$ with its bi-invariant metric and $R^n$ the euclidean space of dimension $n$. What is the minimal value of $n$ such that there exists an isometric embedding $f: SO(3) \to R^n$?
- 251
21
votes
1
answer
1k
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A differentiable isometry is smooth?
I posted this question in MSE but got no response (even after giving a bounty), so I am trying here.
Let $M,N$ be smooth $d$-dimensional Riemannian manifolds.
Suppose $f:M \to N$ is a differentiable ...
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17
votes
2
answers
1k
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Are there some intrinsic invariants of surfaces other than Gaussian curvature?
The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$.
Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically ...
- 366
16
votes
1
answer
601
views
If all balls around two points are isometric... -- manifold version
This question is a natural follow-up of this other question, asked earlier today by wspin.
Let's say that a metric space $(X,d)$ has two poles if:
there are two distinct points $x$, $y$ such that ...
- 10.9k
13
votes
3
answers
2k
views
Isometry group of a compact hyperbolic surface
Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have non-...
- 133
13
votes
3
answers
2k
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Is there a global obstruction for a diffeomorphism to be an isometry?
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 $T$...
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10
votes
4
answers
711
views
Palais's and Kobayashi's theorems on automorphism groups of geometric structures
My question concerns two results in the neighborhood of the standard theorem of Myers-Steenrod that isometry groups of Riemannian manifolds are Lie groups. Both appear in the first chapter of ...
- 486
7
votes
1
answer
373
views
Are metric isometries smooth at the boundary?
Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth).
Suppose we have a map $\phi:M \to N$ which satisfies the following properties:
$$(1) \, \,...
- 6,741