All Questions
Tagged with dg.differential-geometry isometries
8 questions
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
17
votes
2
answers
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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
13
votes
3
answers
2k
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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 $T$...
- 6,741
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
6
votes
1
answer
1k
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Laplace-Beltrami and the isometry group
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 ...
5
votes
2
answers
2k
views
Isometric embeddings of metric spaces in Hilbert spaces
There are plenty of isometric embeddings of metric spaces in Banach spaces. Nevertheless, I have been unable to find any significant result on isometric embeddings into Hilbert spaces. My question is: ...
- 5,407