All Questions
Tagged with lie-groups smooth-manifolds
11 questions
1
vote
1
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345
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Is the manifold of complex points of a quotient of compact groups just the tangent bundle?
In great generality a Lie group mod its maximal compact subgroup is contractible (for example this is true for all connected Lie groups). Whenever this is true then the Lie group $ D $ is ...
- 4,081
41
votes
3
answers
3k
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What is the classifying space of "G-bundles with connections"
Let $G$ be a (maybe Lie) group, and $M$ a space (perhaps a manifold). Then a principal $G$-bundle over $M$ is a bundle $P \to M$ on which $G$ acts (by fiber-preserving maps), so that each fiber is a $...
- 54.6k
12
votes
3
answers
3k
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Non-Lie Subgroups
A result of Borel and Lichnerowicz states that the holonomy group of a connection on a principal $G$-bundle is a Lie subgroup of $G$ (Cartan had earlier asserted this, but apparently without proof).
...
- 2,806
8
votes
1
answer
599
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Exact condition for smooth homogeneous to imply Riemannian homogeneous for compact manifolds
Let $ (M,g) $ be a homogeneous Riemannian manifold. That is, the isometry group $ Iso(M,g) $ acts transitively on $ M $. Let $ \pi_1(M) $ be the fundamental group of $ M $. Then $ \pi_1(M) $ has ...
- 4,081
6
votes
1
answer
1k
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Does a Trivial Tangent Bundle Induce a Multiplication?
Let $M$ be a connected smooth manifold, and assume that it is parallelisable; that is, its tangent bundle is trivial. Does $M$ admit an H space structure? That is, does there exist a smooth map $\mu:...
- 208
5
votes
1
answer
355
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Nonlinear sigma models with non-compact groups / target spaces
A nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T.
The target manifold T is equipped with a Riemannian metric g. Σ is ...
- 10.5k
5
votes
1
answer
720
views
Poincaré–Bendixson theorem on the torus
I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem:
THEOREM. Let $M$ be a ...
- 201
5
votes
0
answers
135
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Specify the embedding of Lie groups (via the representation map) precisely as the embedding of two differentiable manifolds
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
- 10.5k
4
votes
1
answer
236
views
Submanifold of a Lie group whose tangent bundle is invariant under group (left) action
Edit: According to the interesting comment of Tobias Fritz we revise the question.
Assume that $G$ is a Lie group and $M\subseteq G$ is a closed connected smooth submanifold of $G$ containing the ...
- 356
3
votes
0
answers
865
views
Non-Abelian fundamental group? --- a bizarre example
For the quotient space $G=G_0/G_1$, knowing the homotopy
groups of $G_0$ and $G_1$, one can determine homotopy groups from the long
exact sequence
$$
...
\to \pi_n(G_1) \to \pi_n(G_0) \to \...
- 10.5k
3
votes
2
answers
2k
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Is a manifold paracompact? Should it be?
We will say that a Hausdorff topological space $X$ is a smooth manifold if there is an open cover $(U_{\alpha})$ of $X$ and a corresponding collection of homeomorphisms $\varphi_{\alpha} : U_{\alpha} \...
- 1,379