All Questions
Tagged with lie-groups dg.differential-geometry
567 questions
6
votes
4
answers
889
views
On the determination of a quadratic form from its isotropy group
Let $F:\mathbf{R}^n\rightarrow\mathbf{R}$ be a non-degenerate quadratic forms. Let
$$
O(F):=\{g\in GL_n(\mathbf{R}):F(gv)=F(v),\forall v\in \mathbf{R}^n\}
$$
be the isotropy group of $F$.
Q: So how ...
- 7,609
6
votes
5
answers
3k
views
Lie group operation and tangent vectors
Suppose we have two differentiable paths $\alpha$ and $\beta$ thru the identity of a Lie group $G$, $\alpha(0)=\beta(0)=e$ the identity element. Denote $\alpha\beta$ the path given by $\alpha\beta(t)=\...
- 63
6
votes
5
answers
2k
views
How to characterize Dirac's gamma matrices in differential geometry?
I want to understand what is the interpretation of Dirac gamma matrices in differential geometry. Basically, I am considering the Dirac matrices as 3-indexed tensors, which means a tensor with 1 ...
- 161
6
votes
2
answers
1k
views
Parallel forms and cohomology of symmetric spaces
Let $G/H$ be a compact symmetric space. Then I believe the following is true: if $\alpha \in \Omega^k(G/H)$ and $\nabla$ the Levi-Civita connection, then
$$
(\alpha \text{ is induced by an $\...
- 3,244
6
votes
1
answer
1k
views
Todd class and Baker-Campbell-Hausdorff, or the curious number $12$
The number $12$ (or, probably we shall say Bernoulli numbers in general) appears in many places in Mathematics, sometimes leading to unexpected connections between different topics.
For instance, ...
- 1,525
6
votes
1
answer
864
views
Kähler form on complex Lie group
Hallo,
Let $G$ be a semi-simple, compact Lie Group. Consider its complexification $G_{\mathbb{C}}$. Does there exist a Kähler structure on $G_{\mathbb{C}}$ which is $G$-invariant (maybe in a ...
- 339
6
votes
2
answers
501
views
Group of diffeomorphisms and its tangent space i.e. its Lie algebra
So I feel like there are many questions and also many sources on what I am asking, but I still don't understand what I think is a very basic thing in my head:
It is known, that for a Lie group $G$ (...
- 63
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
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 ...
- 3,541
6
votes
1
answer
323
views
Deformations of the 4-sphere with 8-dimensional isometry groups
I am looking for deformations of the 4-sphere with 8-dimensional isometry group, like a 4-dimensional Berger sphere.
- 427
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
6
votes
1
answer
256
views
Geodesic in space of circulant matrices
I'm trying to find the geodesic that connects the identity with some circulant, symmetric matrix $U\in\mathrm{GL}(N,\mathbb{R})$, meaning we have
\begin{align}
U=\left(\begin{array}{ccc}
u_1 & ...
- 285
6
votes
4
answers
331
views
How can one show $G/T$ is a coadjoint orbit for a compact Lie group $G$ and $T$ its maximal torus?
$\newcommand{\g}{\mathfrak{g}}$Let $G$ be a compact Lie group and $\g$ its Lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) can be identified to a ...
- 61
6
votes
1
answer
930
views
Uniform lattices in semisimple Lie groups
Let $\Gamma$ be a uniform lattice in a semisimple Lie group $G$.
Must $\Gamma$ be virtually torsion-free?
If (1) is true, then does this work more generally if $G$ is reductive?
I am motivated by a ...
- 687
6
votes
1
answer
466
views
Hyperbolic manifolds with infinite cyclic fundamental group
It is a well known fact that there is a correspondence between complete hyperbolic $n$-manifolds up to isometry and discrete subgroups of isometries of the hyperbolic space $\mathbb{H}^n$ that act ...
- 2,533
6
votes
1
answer
369
views
Can I bring the Kirillov 2-form on coadjoint orbits to adjoint orbits?
I tried asking this question on stackexchange and received no response.
Given a semisimple Lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirillov 2-form. Can I ...
user125834
6
votes
1
answer
3k
views
Difference between the Laplacian and the sub-Laplacian of a Lie group
Given a Lie group $G$, what is the difference between the Laplacian $\Delta$ and the sub-Laplacian $\Delta_{sub}$ of $G$. And what are the properties that we lose when going from sub-Laplace to ...
- 650
6
votes
2
answers
729
views
Relationship between the Lie functor applied to a Lie group action, and the fundamental vector field mapping?
Let $M$ be a smooth manifold, and $G$ a Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra of the diffeomorphism group of $M$ is the Lie algebra of vector fields on $M$; that is $\text{Lie}(\...
- 6,025
6
votes
1
answer
495
views
geodesic of $\rm SO(3)$ as a compact Lie group vs as a Riemannian symmetric space
I got a little bit confused about the definition of geodesic for $\rm SO(3)$ as
a compact Lie group
a Riemannian symmetric space
In the former case, it is given by the usual matrix exponential:
$$
\...
- 163
6
votes
1
answer
644
views
Torus bundles and compact solvmanifolds
I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here.
Let
$$
T^n \to M \to T^m ...
- 4,081
6
votes
1
answer
390
views
Equivariant implicit function theorem
Let $f:\mathbb{R}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a smooth function and $G\subset \operatorname{SO}(n)$ be a $1$-dimensional compact Lie group (diffeomorphic to the circle). ...
- 63
6
votes
2
answers
379
views
About Lie group $G$ has this escape property？
Every Lie group $G$ has the following escape property: For every $x \ne e$ in a sufficiently small neighborhood $U$ of the identity $e$ in $G$, there is a integer $n$ such that $x^n$ is not in $U$.
...
- 71
6
votes
1
answer
596
views
Vector fields, diffeomorphism subgroups and lie group actions
Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization:
Let $\{X_j\} \in Vect(M)$ be a ...
- 7,789
6
votes
1
answer
645
views
The group of isometries of Shahshahani metric
Edit: 28 January 2023 I just realized that this metric is frequently used in this paper
https://hal.science/hal-01382281/document
Let $$M=\{(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\mid x_i>0,\;i=1,2,\...
- 356
6
votes
1
answer
1k
views
An integral with respect to the Haar measure on a unitary group
Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...
- 141
6
votes
1
answer
325
views
An extension of symplectomorphism group
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$Let $\omega=\sum dx_i\wedge dy_i$ be the standard symplectic structure of $\mathbb{R}^{2n}=\mathbb{R}^{n}\times \mathbb{R}^n$.
We consider the ...
- 356
6
votes
1
answer
213
views
When are principal bundles supporting Cartan connections isomorphic?
Suppose I have two Cartan geometries $(\mathscr{G}_1,\omega_1)$ and $(\mathscr{G}_2,\omega_2)$ of type $(G,H)$ over the same manifold $M$. What conditions on $G$ and $H$ allow us to conclude that $\...
- 779
6
votes
1
answer
213
views
On the orbit of a Fréchet Lie group action
Suppose that $G$ is a Fréchet Lie group acting on a Fréchet manifold $X$.
Fix $x\in X$ and let $\alpha(t)$ be a smooth path in $X$ such that
$$
\begin{cases}
\alpha(0)=x\\
\alpha(t)\in G\cdot x.
\end{...
- 305
6
votes
1
answer
1k
views
Laplace-Beltrami operator on a Lie group
For an arbitrary Lie group, is it always possible to chose a left-invariant Riemannian metric such that the Laplace-Beltrami operator $\Delta$ is given by
$$\Delta f = \delta^{i j} X_i X_j f$$
for ...
- 6,025
6
votes
2
answers
426
views
Approximating the action of the U(N) exponential map
Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group:
$$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$
Here, $H$ is an NxN skew-Hermitian matrix (for very ...
- 161
6
votes
1
answer
977
views
reference for the slice theorem for Banach Lie group actions on Banach manifolds
I am looking for a reference treating the slice theorem for Banach Lie group actions on Banach manifolds, i.e. proving that a smooth, free and proper action of a Banach Lie group $G$ on a Banach ...
- 2,935
6
votes
0
answers
147
views
Maximum symmetry metric on irreducible compact symmetric space
Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. ...
- 4,081
6
votes
0
answers
177
views
Equivariant Morse theory for non-compact Lie groups
Let $G$ be a Lie group acting properly on a smooth manifold $M$. The (non-equivariant) definition of a Morse function does not carry over to equivariant functions $M \rightarrow \mathbb{R}$ (where $\...
- 198
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 ...
- 779
6
votes
0
answers
304
views
Group Representations and Holomorphic Vectors Bundles over Homogeneous Spaces (extending Borel--Weil)
For a flag manifold $F$ of a group $G$, the Borel--Weil theorem deals with representations of $G$ on the holomorphic sections of the line bundles over $F$.
Let us consider a general framework than ...
- 3,399
5
votes
2
answers
733
views
Are maximal connected semisimple subgroups automatically closed?
(Yet another question in a series demonstrating my rather embarrassing ignorance of standard Lie theory... I hope this is not too basic for MO!)
To be a little more precise: let $G$ be a real ...
- 25.8k
5
votes
2
answers
480
views
Maximum symmetry metric on $ \mathbb{C}P^n $
Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. ...
- 4,081
5
votes
3
answers
712
views
A followup on non-homogeneous spaces.
This question asks for an example of a manifold which is not a homogeneous space of any Lie Group, and many examples are given in the answers. However: is there a an example known with a metric of ...
- 96.4k
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
5
votes
1
answer
261
views
Non-integrable almost complex structure for complex projective $3$-space
It is well known that complex projective three space $\mathbb{C}\mathbf{P}^3$ is a complex manifold. However it also possess a non-integrable almost-complex structure (as discussed in this article for ...
5
votes
1
answer
1k
views
What are some interesting examples of quotients by Lie group actions?
I’ve been going through Lee’s Introduction to Differential Geometry. It’s a great book but I feel it lacks examples and concrete applications of the ideas presented. I want to work out a list of ...
5
votes
1
answer
852
views
Normal forms for homogeneous cubic polynomials in $\mathbb{R}[x_1, x_2, x_3]$
Is there a standard normal form for homogeneous cubic polynomials in $\mathbb{R}[x_1, x_2, x_3]$? Or, put another way, is there a nice way to describe the orbit space of the natural (diagonal) action ...
- 818
5
votes
1
answer
252
views
Does a compact Lie group action on a family of compact manifolds have diffeomorphic fixed point submanifolds?
Let $\pi: M\to B$ be a fiber bundle of smooth manifolds with $B$ connected and each fiber of $\pi$ is a compact manifold. Let $G$ be a compact Lie group acting smoothly on $M$ such that
$\pi(g\cdot m)=...
- 9,019
5
votes
2
answers
733
views
Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?
Let $(M,g)$ be a (connected, paracompact, $C^{\infty}$-smooth) Riemannian manifold with Riemannian metric $g$. The exponential map is defined for each point $p \in M$ to be the map $\exp_p : T_p M \to ...
- 1,379
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 ...
5
votes
1
answer
226
views
Nontrivial extension of the action of complex hyperbolic group $H$ on $\mathbb{C}$
Inspired by this question about conjugation of reql analytic maps to a holomorphic function and with a group action view point we ask the following question.
The complex Lie group $H=\...
- 356
5
votes
1
answer
303
views
Iwasawa decomposition of a non-compact semisimple Lie group?
A (non-compact) symmetric space is of the form $G/K$, where $G$ is a (non-compact) semisimple Lie group defined over $\mathbb{R}$, and $K$ is a maximal compact subgroup of $G$.
Let $M = G/K$ be a rank-...
- 650
5
votes
1
answer
978
views
Existence proof of Bourbaki, Differentiable and Analytic Manifolds
I am reading through Chapter III of Bourbaki, Lie Groups and Lie Algebras, and many proofs cite the Bourbaki volume Differentiable and Analytic Manifolds. I can't find this book anywhere. Does it ...
- 6,180
5
votes
1
answer
243
views
Triviality of a fiber bundle
Is the principal fiber bundle $GL^+(6,\mathbb R)$ over $GL^+(6,\mathbb R)/SL(3,\mathbb C)$ trivial ?
- 79
5
votes
1
answer
269
views
Normal form for trace-free real cubic forms in 3 variables under SO(3)-action?
I'm looking at irreducible, real representations of $SO(3)$. The 5-dimensional irrep is isomorphic to the space of trace-free quadratic forms on $\mathbb{R}^3$, and we all know that any such ...
- 818