All Questions
18 questions
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
18
votes
1
answer
980
views
Possible isometries of a positively curved $S^2\times S^2$
Just to put things in perspective, recall that the Hopf Conjecture asks whether $S^2\times S^2$ admits a metric of positive sectional curvature. By the work of Hsiang-Kleiner, it is known that, if $S^...
- 5,891
17
votes
2
answers
1k
views
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
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
views
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
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
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
8
votes
3
answers
629
views
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,...
- 5,349
7
votes
1
answer
1k
views
The surjectivity of the exponential map for the isometry group
Little is known on general conditions guaranteeing that the exponential map between a Lie algebra and an associated Lie group is surjective.
Let $M$ be a noncompact connected Riemann manifold, and $G$...
- 5,407
7
votes
1
answer
497
views
Open problems about CMC hypersurfaces with symmetries?
Recently, Andrews and Li announced a complete classification of CMC ($H=const.$) tori in $S^3$, confirming a conjecture of Pinkall and Sterling. Their main result is that any such torus is ...
- 5,891
6
votes
1
answer
734
views
Are the Sasaki metrics on tangent and cotangent bundle isomorphic?
Let $(M,g)$ be a Riemannian manifold. Then there is the well-known
Sasaki metric that makes $(TM,\hat{g})$ a Riemannian manifold. In a
similar way, one can construct a Sasaki metric $\bar{g}$ on the
...
- 2,481
6
votes
1
answer
370
views
Does the isometry group determine the Riemannian metric?
Suppose $G \subset \text{Iso}(M)$ is a Lie group acting smoothly on a (pseudo-)Riemannian manifold $(M, g)$. Then $G$ induces a Lie algebra of Killing vectors on $M$. In this paper by Goenner and ...
- 203
6
votes
0
answers
691
views
Isometries of Compact Semisimple Lie Groups
In this delightful question, the poster mentioned that the isometry group of a compact Lie group $G$, equipped with the metric from the Killing form, is $G\times G/Z(G)$, where $Z(G)$ is the center of ...
- 779
5
votes
4
answers
2k
views
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 ...
- 3,541
4
votes
1
answer
254
views
Isometry group of a left-invariant Riemannian metric on $\mathrm{SU}(2)$
Recall that
\begin{equation}
\mathbb{S}^3=\operatorname{SU}(2)=\left\{
\begin{pmatrix}
z&w\\
-\bar{w}&\bar{z}
\end{pmatrix}
,|z|^2+|w|^2=1
\right\}
\end{...
- 1,441
2
votes
1
answer
261
views
isometric action on the $n$-sphere
Let $S^n$ be the $n$-sphere. If $n=2k+1$ is odd, then we can identify $S^n$ as a subset of $\mathbb{C}^{k+1}$. We define the $S^1$ action on $S^n$ by multiplication, namely
$$ \Psi \colon S^1 \times ...
- 501
2
votes
1
answer
101
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 ...
- 1,467
1
vote
1
answer
207
views
Riemann isometry vs Euclidean bi-Lipschitz mapping
Assume that $\gamma$ is a rectifiable Jordan curve in the complex plane of length $2\pi$. Then there exists a Riemann isometry $f$ between $\gamma$ and the unit circle $T$. My question is, does this ...
- 303