All Questions
91 questions
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 ...
- 858
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 ...
CommunityBot
- 1
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 ...
- 63.9k
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
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 ...
- 28k
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
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 ...
- 31.5k
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 ...
- 28.9k
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 ...
- 28.9k
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 ...
- 28.9k
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 (...
- 108k
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
9
votes
2
answers
638
views
Curvature of the Cayley projective plane
The Cayley projective plane can be realized as the compact homogeneous space $F_4/\mathrm{Spin}(9)$. In this way one can compute the curvature of this symmetric space in terms of a suitable ...
- 63.9k
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 ...
- 13.6k
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 ...
- 496
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
1
answer
274
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}^...
- 1,624
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 ...
- 6,813
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 ...
CommunityBot
- 1
16
votes
1
answer
2k
views
A careful roadtrip from locally symmetric spaces to algebra
I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning completely)...
- 7,789
3
votes
2
answers
1k
views
Reference for homogeneous spaces
I am a graduate student of differential geometry.
I would like to get an overview over the way, how results are usually obtained for homogeneous spaces by Lie algebraic methods. By definition a ...
- 26.3k
3
votes
1
answer
116
views
Estimates for radii of slices for proper Lie group actions
Let $G$ be a Lie group acting properly on a smooth manifold $M$, and equip $M$ with a Riemannian manifold that is adapted to the foliation by orbits. The celebrated theorem of Palais is that there ...
- 25.3k
20
votes
3
answers
9k
views
Curvature of a Lie group
Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
- 23.3k
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 ...
- 63.9k
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 ...
CommunityBot
- 1
1
vote
0
answers
242
views
Dimension of tangent space to manifold of cross section slices
Given a function $\Phi:\Omega^{\Phi}\subset \mathbb{R}^3\rightarrow\mathbb{R}$, we intruduce its planar cross section slices $\phi^{s}:\Omega^{\phi}\subset \mathbb{R}^2\rightarrow\mathbb{R}$, using a ...
- 111
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 ...
CommunityBot
- 1
4
votes
0
answers
335
views
The geometry of the holomorph of a Lie group
Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)
Is Hol$(G)$ always a Lie group?
If the answer is yes our main questions:
1.For a left invariant ...
CommunityBot
- 1
6
votes
0
answers
690
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 ...
CommunityBot
- 1
6
votes
1
answer
1k
views
Laplace-Beltrami operator on a Lie group
For an arbitrary Lie group, is it always possible to chose a left-invariant Riemannian metric such that the Laplace-Beltrami operator $\Delta$ is given by
$$\Delta f = \delta^{i j} X_i X_j f$$
for ...
- 6,025
7
votes
1
answer
1k
views
The surjectivity of the exponential map for the isometry group
Little is known on general conditions guaranteeing that the exponential map between a Lie algebra and an associated Lie group is surjective.
Let $M$ be a noncompact connected Riemann manifold, and $G$...
- 108k
35
votes
5
answers
4k
views
$G_2$ and Geometry
In a recent question Deane Yang mentioned the beautiful Riemannian geometry that comes up when looking at $G_2$. I am wondering if people could expand on the geometry related to the exceptional Lie ...
CommunityBot
- 1
7
votes
3
answers
1k
views
Is the group of isometries of a homogeneous Riemannian manifold maximal?
I have a homogeneous Riemannian manifold X with isometry group Iso. Is Iso a maximal group? By maximal group, I mean that there does not exist another group G such that:
Iso is a proper subgroup of G,...
CommunityBot
- 1
1
vote
1
answer
215
views
Relation between volume of reduced space and phase space
Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is $$...
- 27.8k
1
vote
1
answer
340
views
Riemannian metric and Volume form for $SE(n)$ and/or $E(n)$
I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the (left ...
- 3,388
2
votes
1
answer
283
views
Isometric embedding of a compact Lie Group in $M(n,\mathbb{C})$
Greetings,
Let $G$ be a compact Lie group with a bi-invariant inner product $h$ on it. Can one embedd $G$ in $M(n,\mathbb{C})$ isometrically for some $n \in \mathbb{N}$. By isometrically I mean that ...
- 25.3k
15
votes
0
answers
637
views
"Homogeneity" of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]
My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...
- 5,891
6
votes
2
answers
903
views
Ricci curvature of the symplectic group
Is the Ricci curvature of the compact symplectic group $Sp(n)$ bounded below by $cn$ for some constant $c > 0$ independent of $n$?
For $O(n)$ and $U(n)$ I know many references which state such a ...
- 11.4k
3
votes
2
answers
1k
views
What are the computationally useful ways of thinking about Killing fields?
One definition of the Killing field is as those vector fields along which the Lie Derivative of the metric vanishes. But for very many calculation purposes the useful way to think of them when dealing ...
CommunityBot
- 1
2
votes
1
answer
826
views
Frobenius Theorem
Say a manifold M has 3 vector fields S,T and R whose Lie brackets satisfy the equations $[S,T]=R$, $[R,S]=T$ and $[T,R]=S$
Then I suppose the following properties hold for M,
There exists a metric ...
- 45k
6
votes
4
answers
3k
views
Killing fields on homogeneous spaces
Let $G$ be a compact lie group and $H$ a closed subgroup and hence think of $G/H$ as a homogeneous space.
Then how are the Killing fields on $G/H$ the projection of the right-invariant vector fields ...
- 27.5k