All Questions
16 questions
5
votes
1
answer
530
views
Geodesic distance on $\mathrm{SO}(n)$
$\DeclareMathOperator\SO{SO}$Recently I came across this old MSE post or this paper (w.o. proof) discussing the geodesic distance on $\SO(n)$ when it is equipped with the left-invariant Riemannian ...
- 364
1
vote
0
answers
70
views
Orbit projection geometry
Background:
As shown in [1] and [2], for a closed smooth submanifold $M$ of $\mathbb R^d$, the domain $D_M$ of the projection map $P_M:D_M\rightarrow M$ has a dense interior $\Omega_M$ over which $P_M|...
- 71
2
votes
1
answer
325
views
Orbit space of $\mathrm{SO}(3)$ irreducible representations
$\DeclareMathOperator\SO{SO}$Consider the $7$-dimensional $\mathbb R^7$ real irreducible orthogonal representation of $\SO(3)$. I am seeking a description of the orbit space (when the action is ...
- 71
7
votes
1
answer
367
views
Does complexified isometry group act transitively on tangent bundle of compact Riemannian manifold?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$Let $ g $ be the round metric on the sphere $ S^n $. Since $ S^...
- 4,081
4
votes
1
answer
230
views
Generalizing a result about hyperbolic 2-folds to hyperbolic 3-folds
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$Let $ \Sigma_g $ be a compact orientable surface of genus $ g $. Let the subgroup $ \pi_1(\Sigma) $ of $ \SL_2(\...
- 4,081
1
vote
1
answer
319
views
Are there any applications of linear algebra over the complex numbers, where the role of complex conjugation is replaced with the trivial involution?
The complex inner product $\langle u, v \rangle$ is a special case of a sesquilinear form over a field. Its definition is $\langle u, v \rangle = \sum_{i} u_i \overline{v_i}$. There is clearly the ...
- 4,943
3
votes
0
answers
144
views
Erlangen program for "network geometry"
The subject of network geometry (Boguna et al., Network Geometry,
Nature Reviews Physics 2021) looks at "geometric aspects" of complex networks.
This is about studying a metric on the nodes, ...
- 640
18
votes
1
answer
2k
views
The group of isometries of a manifold is a Lie group, isn't it?
Let $M$ be a connected finite dimensional topological manifold and $g$ be any metric on it that induces the topology of $M$ ($g$ is not a Riemannian metric). How to prove that the group of isometries ...
- 14.3k
-1
votes
1
answer
181
views
Reparameterization and group structure
I ran into the following question; let $x,y$ be two points in $\mathbb{R}^d$. Let $(\psi_t)_{t\geq 0}$ be the mapping from $\mathbb{R}^{2d}$ to $\mathbb{R}^{2d}$ defined, for all $t\geq 0$, by
$$
\...
user153086
9
votes
1
answer
255
views
On the diameter of left-invariant sub-Riemannian structures on a compact Lie group
Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ of dimension $m$.
We fix an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak g$.
We may assume (in case is necessary) ...
- 2,446
16
votes
2
answers
756
views
Can a sphere glued into a soft 3d-mattress rotate continuously? (manifolds, SU(2) and the belt trick)
The question is triggered by the wonderful animations by Jason Hise:
https://www.youtube.com/watch?v=LLw3BaliDUQ
https://www.youtube.com/watch?v=6Ul_-ABYaYU
https://www.youtube.com/watch?v=...
5
votes
2
answers
359
views
References for metrics in matrix groups
I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...
11
votes
4
answers
369
views
Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant
The Lie group $GL(n)$ being a manifold is locally path-connected. Consider its connected component of the identity $C\subseteq\mathbb{R}^{n^2}$. What is a good way of showing that $C$ is a tame ...
- 16.6k
11
votes
1
answer
726
views
Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$?
$\newcommand{\til}{\tilde}$
Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds.
Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open ...
- 6,741
1
vote
1
answer
214
views
Orbits of Product Lie Groups Action
Hi to all,
Let $G$ be a Lie group of linear isometries of $\mathbb{R}^n_{\nu}$ ($\mathbb{R}^n_{\nu}$ is the semi-Euclidean space) and $G_1$ ,$G_2$ two Lie subgroups of $G$. Let $G_1 \times G_2$ as ...
user30435
2
votes
1
answer
268
views
Riemannian Hausdorff distance between two conjugacy classes in a compact Lie group
I am interested in the distance between two conjugacy classes in a group like $SO(n)$. However let's consider $U(n)$ for simplicity. My conjecture is that the Hausdorff distance between the conjugacy ...
- 4,466