All Questions
Tagged with riemannian-geometry lie-groups
155 questions
2
votes
0
answers
54
views
Want a minimal subgroup, whose orbits cover a submanifold, that is contained in a maximal subgroup which leaves the submanifold invariant
Say we have a Lie group $G$ acting transitively on smooth manifold $M$ and take a submanifold $S\subseteq M$. It seems to me that there should be some minimal subgroup $G'\subseteq G$ such that $S\...
- 21
1
vote
0
answers
134
views
Hausdorff dimension of a compact Lie group [closed]
Let $G$ be a compact Lie group (for simplicity assume $G=SO(3)$). Equip $G$ with a left-invariant Riemannian metric and let $m$ be left-invariant Haar measure on $G$.
Now that $G$ is a metric space ...
- 323
3
votes
0
answers
142
views
Homogeneous Carnot group, its Lie algebra and Carnot-Carathéodory ball
Background: Let the smooth vector fields $X=(X_1,\cdots,X_m)$ define on $\mathbb{R}^n$ and they satisfy the following assumption:
(H1): There is a dilation structure
$$\delta_{t}:\mathbb{R}^n\to \...
- 561
9
votes
0
answers
326
views
Maximal geodesic spheres in the "octooctonic projective plane"
Boris Rosenfeld claimed that the 128-dimensional compact Riemannian symmetric space on which $\mathrm{E}_8$ acts as isometries could be seen as the "octooctonionic projective plane", $(\...
- 22.3k
-1
votes
1
answer
230
views
Orthonormal frame on $\mathbb{S}^3$ orthogonal to foliations [duplicate]
Does there exist a smooth orthonormal frame $X_1,X_2,X_3$ on $\mathbb{S}^3$ such that the distribution spanned by $X_i$ and $X_j$ is integrable for all $1\leq i,j\leq 3$?
- 89
5
votes
1
answer
206
views
An explicit description of $\operatorname{Isom}(\widetilde{\operatorname{Sl}_2})$
$\DeclareMathOperator\Sl{Sl}\DeclareMathOperator\PSl{PSl}\DeclareMathOperator\Isom{Isom}$Let $\widetilde{\Sl_2}$ be the Thurson geometry that can either be described as the universal cover of $\PSl(2,\...
- 223
8
votes
1
answer
610
views
Are invariant forms on homogeneous spaces necessarily closed?
Take a compact homogeneous space $G/K$, and a left $G$-invariant differential $k$-form $\omega \in \Omega^k(G/K)$. Will $\omega$ necessarily be closed? Might it even be harmonic when $G/K$ is endowed ...
- 141
1
vote
1
answer
96
views
Small deformations of maximally symmetric 3-spaces
I am looking for all 'small' deformations of the three 3-dimensional Riemannian spaces with maximal symmetry, the pseudo-sphere, Eucidean space and the sphere. By a theorem by Fubini (1903) the ...
- 427
2
votes
0
answers
170
views
Totally geodesic submanifolds of SO(3) [closed]
Consider the special orthogonal group $SO(3)$ with its bi-invariant metric (or equivalently, with the metric induced by its standard embedding to the space of $3\times 3$ real matrices).
Obviously, $...
- 381
3
votes
1
answer
309
views
Continuity/Lipschitz regularity of exponential map from $C_c$ to $\operatorname{Diff}_c$?
For finite-dimensional Lie algebras, see this for a nice example, the exponential map is smooth and in particular, it is locally-Lipschitz onto its image. However, things are different when moving to ...
- 5,405
1
vote
0
answers
517
views
Horizontal lift of fundamental vector field
Suppose $\theta\colon G\times M\to M$ is a transitive smooth left action of a compact Lie group $G$ on a manifold $M$ and $\pi\colon G\to M\cong G/K$ the corresponding smooth submersion for some ...
- 193
-1
votes
1
answer
97
views
A manifold or Riemannian structure on the space of all conjugacy classes of a compact Lie group [closed]
Let $G$ be a compact Lie group.
Is each conjugacy class a closed subset of $G$?
Define the conjugacy equivalent relation $g\sim h$ if $g$ is conjugate to $h$.Is $G/\sim$ a Haussdoef space with ...
- 356
3
votes
1
answer
347
views
On functional equation $f\circ \exp=\exp \circ Df$ on a Riemannian manifold or a Lie Group
Let $M$ be a Riemannian manifold or a Lie group whose corresponding exp map (in corresponding context) is denoted by "exp" which is a map $\exp:TM\to M$
We search for the set $\mathcal{H}...
- 356
1
vote
0
answers
80
views
Quotient of Euclidean space with maximal volume growth
Let $\Gamma$ be a discrete subgroup of the isometry group of $\mathbb R^n$ and $O=\mathbb R^n/\Gamma$ is the orbifold.
If there exists a point $p \in O$ such that
$$
\lim_{r \to \infty}\frac{\text{...
- 2,535
7
votes
1
answer
1k
views
Help with definition of Liouville measure
$\require{AMScd}$For a Riemannian manifold $M$, I have read authors talking about a 'Liouville measure' on the unit tangent bundle $\operatorname{T}^1(M)$ and then proceed to claim/prove that it is ...
- 675
2
votes
1
answer
328
views
Totally geodesic submanifolds of bi-invariant Lie groups
Let $G$ be a Lie group equipped with a bi-invariant metric. I have some questions concerning the totally geodesic and the flat (all sectional curvatures zero) submanifolds of $G$.
I known that every ...
- 965
4
votes
3
answers
862
views
Is every homogeneous space Riemannian homogeneous?
A manifold $M$ together with a transitive $G$-action is always diffeomorphic a quotient $G/H$ for $H < G$ Lie groups. On the other hand, there might be a proper subgroup of $G$ that also acts ...
- 177
2
votes
0
answers
135
views
Left invariant Haar measure of $\operatorname{Isom}(\mathbb H^n)$ and finite volume subsets of $\mathbb H^n$
(I have migrated this question from math.SE)
The isometry groups of the Riemannian manifolds $\mathbb E^n, \mathbb H^n$ and $\mathbb S^n$ all have Lie groups structures, so if $X$ is any of these ...
3
votes
3
answers
526
views
3-dimensional Riemannian manifolds with 4-dimensional isometry group
Is there a list of all 3-dimensional, connected Riemannian manifolds with 4-dimensional isometry group?
- 427
2
votes
0
answers
260
views
A geometric property of certain Lie groups
I call Poincaré $n$-half-space group the semidirect product of $\mathbb{R}^{n-1}$ and $\mathbb{R}^+$, where the action is by homotheties; equivalently as the group of translations and positive ...
- 356
4
votes
0
answers
114
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Representation theoretic characterisation of symmetric spaces
Let $G$ be a simple compact Lie group and $H$ a closed subgroup.
Let $\mathfrak{h}\subset \mathfrak{g}$ denote the corresponding Lie algebras. Let $\mathfrak{m}$ be an orthogonal complement to $\...
- 81
2
votes
1
answer
86
views
Compact group actions with uniformly bounded derivatives
Suppose we have a smooth action of a compact Lie group $G$ on a non-compact smooth manifold $M$, denoted by
$$\phi:G\times M\rightarrow M.$$
Differentiating $\phi$ at a point $x\in M$ gives a map that ...
- 1,903
3
votes
0
answers
60
views
Transformation between nearby tangent planes [closed]
This question is kinda long, but the picture is quite clear.
Question: Let $(M,g)$ be a Riemannian manifold, $p$ a point on $M$, $U$ an open neighborhood of $0\in T_pM$ such that $exp_p|_U$ is a ...
- 5,230
5
votes
1
answer
549
views
Volume of balls in homogeneous manifolds
Let $X=G/H$ be a homogeneous manifold, where $G$ and $H$ are connected Lie groups and assume there is given a $G$-invariant Riemannian metric on $X$.
Let $B(R)$ be the closed ball of radius $R>0$ ...
user130903
5
votes
2
answers
377
views
Existence of an isotopy in Riemannian manifold
Let $(M,g)$ be a Riemannian manifold, and $p,q\in M$ be two fixed points. We assume $p,q$ are close enough. Say, we assume $p$ and $q$ are in the same normal coordinate chart. It is clear that there ...
- 2,789
4
votes
0
answers
244
views
Gram-Schmidt map as a Riemannian submersion
We equip $\mathrm{GL}(n,\mathbb{R})$ and $\mathrm{O}(n)$ with their left-invariant metrics, whose restrictions to the corresponding neutral elements is the standard inner product $\mathrm{Trace}(AB^{\...
- 356
2
votes
0
answers
169
views
More general form of Fourier inversion formula
My question begins as follows: Suppose $G$ is a compact (Lie) group and $A$ is a $G$-module, and denote the action of $G$ on $A$ by $\alpha$. Fix $a\in A$ and view
$$
f:g\mapsto \alpha_g(a)
$$
as an $...
- 21
7
votes
1
answer
546
views
Can a hyperbolic manifold be a product?
I was interested in whether a manifold which admits a metric of constant sectional curvature can be homotopy equivalent to a product of non-contractible manifolds. Of course, there are three cases: ...
- 19.3k
7
votes
2
answers
358
views
Is every Lie subgroup of a Lie group isometric to all its conjugates?
Let $G$ be a Lie group with a left invariant metric. Assume that $N$ is a Lie subgroup of $G$.
For a given $g\in G$, are $N$ and $g^{-1} N g$ necessarily isometric Riemannian manifold when they ...
- 356
4
votes
1
answer
233
views
Flat solvmanifolds?
I was looking for some reference on solvmanifolds and came up with a paper by A. Morgan tilted "The classification of flat solvmanifolds". I know there is a complete classification of flat manifolds ...
- 481
5
votes
1
answer
201
views
The Hausdorff dimension of the union of singular orbits and exceptional orbits
Suppose we have a compact connected Lie group $G$ acting as isometries on a compact manifold $M^n.$ Then is it necessarily true that the Hausdorff dimension of the union of singular and exceptional ...
user78128
8
votes
1
answer
360
views
Lie group actions on $S^n$ with some invariant hypersphere but no totally geodesic ones
Does there exist a compact connected Lie group $G$ acting smoothly as isometries on the standard sphere $S^n$ for some $n\ge 3$, so that no totally geodesic hypersphere $S^{n-1}$ is $G$-invariant, but ...
user78128
3
votes
0
answers
154
views
Classification of Euclidian-like Klein geometries in spirit of Erlangen program
All we know about Erlangen Program approach to geometry. The main idea is to consider Lie Group $G$ acting transitively on smooth manifold $M$ (e.g. nLab). I really like such style of thinking, but ...
- 1,307
6
votes
1
answer
508
views
Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n)
It's well known that the sectional curvatures of a Lie group, endowed with a left-invariant metric have a nice closed-form formula $k(X,Y) = \frac{1}{4} \|[X Y]\|^2$.
I'm wondering if the following (...
- 265
7
votes
1
answer
180
views
existence of riemannian metric on $\text{SL}_3(\mathbb{R})$ with special geodesics
Is there a left-invariant Riemannian metric on $\text{SL}_3(\mathbb{R})$ for which the geodesics (with respect to the corresponding Levi-Civita connection) through the identity are exactly the ...
- 197
6
votes
1
answer
466
views
Holonomy groups of compact Riemannian symmetric spaces
Let $M$ be a compact Riemannian symmetric space. By the classification of Cartan, it belongs to the table of homogeneous spaces given in the Wikipedia page:
https://en.wikipedia.org/wiki/...
- 643
6
votes
1
answer
645
views
The group of isometries of Shahshahani metric
Edit: 28 January 2023 I just realized that this metric is frequently used in this paper
https://hal.science/hal-01382281/document
Let $$M=\{(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\mid x_i>0,\;i=1,2,\...
- 356
8
votes
1
answer
673
views
Classification of compact globally symmetric spaces
It is known that any connected compact Lie group $G$ is a finite quotient of the product of a compact simply connected semisimple Lie group $\tilde{G}$ and a torus $\mathbb{T}^n$ (see for example ...
- 193
2
votes
0
answers
198
views
Left-invariant Riemannian metric on $\text{SL}_3(\mathbb{R})$ under right-translation
Given any left-invariant distance $\text{d}$ on $\text{SL}_3(\mathbb{R})$ (meaning $\text{d}(a,b) = \text{d}(ca,cb)$ for all $a,b,c \in \text{SL}_3(\mathbb{R})$) which is induced by a left-invariant ...
- 197
5
votes
2
answers
662
views
Most natural connection on Lie group: comparison of different pictures
Let $G$ be a Lie group (not necessarily compact). One can equip $G$ with the left invariant metric (or
right invariant but in general there is no biinvariant metric in the noncompact case). Once the ...
- 9,330
7
votes
0
answers
508
views
Explicit formula for the Levi-Civita connection on a non-compact Riemannian symmetric space
Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is ...
- 1,942
9
votes
3
answers
701
views
Diameter of $\mathrm{SU}(2)$ endowed with a left-invariant metric
Basic question: What is the diameter of $\mathrm{SU}(2)$ endowed with a left-invariant metric?
Now, let me give more information.
Set
$$
X_1= \begin{pmatrix} i &\\ &-i \end{pmatrix},\;
X_2= \...
- 2,446
2
votes
0
answers
160
views
When is a simply connected Lie group with an invariant metric of positive sectional curvature compact?
When is a simply connected Lie group with an invariant metric of positive sectional curvature compact?
The point of the question is: does this hypothesis provide a positive lower bound on Ricci ...
- 1,774
2
votes
0
answers
317
views
When is the exponential map injective on generic subspaces?
If you take a flat 2-torus $\mathbb{T}$ and a random 1-dimensional subspace $V \subseteq T_e \mathbb{T}$ (where $e$ is an arbitrary basepoint), then generically the exponential map is injective.
This ...
- 595
2
votes
0
answers
163
views
Geometric and holomorphic structure of $\mathbb{C} \rtimes \mathbb{C} \setminus \{ 0 \}$
Put $G= \mathbb{C} \rtimes_{\phi} \mathbb{C} \setminus \{0\}$ where $\phi_{a} (z)= az$ for $a \in \mathbb{C} \setminus \{0\}$.
$G$ is a real $4$ dimensional Lie group; then it has a unique left ...
- 356
7
votes
2
answers
2k
views
Geodesics equation on Lie groups with left invariant metrics
First of all, I am so sorry if this question is not appropriate to be here. I tried to ask something similar on Math Stack Exchange but it didn't have much attention. Any comment and I delete the ...
- 1,774
2
votes
1
answer
273
views
Isomorphism of Complex Stiefel manifold and Homogeneous space of unitary group, and the Stiefel logarithm problem
It is well known that $U(n)/U(n-k) \cong V_k(\mathbb{C}^n)$ where $U(n)$ is the unitary group, and $V_k(\mathbb{C}^n)$ is the appropriate Stielfel manifold.
I further understand that $V_k(\mathbb{C}^...
- 2,099
4
votes
1
answer
324
views
On the isometry group of a self cartesian product of a Riemannian space
Let $X$ be a complete Riemannian space. Let us denote by $Iso(X)$ the group of isometries of $X$. It is a well-known fact that the group $Iso(X)$, when endowed with the compact-open topology, is a Lie ...
- 7,609
5
votes
1
answer
473
views
Geodesics on Homogeneous Spaces of $SU(n)$
Consider the homogeneous space $SU(n)/K$, where $K$ is a sub-group of $SU(n)$ and the bi-invariant metric on $SU(n)$.
What is the appropriate quotient metric on the homogeneous space and what are the ...
- 2,099
20
votes
1
answer
1k
views
Why does the proof of Myers and Steenrod fail in the Lorentzian case?
This is my first question on this site. I hope it is not inappropriate on MO.
Myers and Steenrod proved 1939 that the isometry group of a Riemannian manifold is a lie group. I add a picture where ...
- 893