All Questions
Tagged with compactness dg.differential-geometry
8 questions
0
votes
0
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155
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Implicit function theorem on curves
I am trying to figure out, whether the IFT can be generalized to curves. Let's say I have a function $G(x,u)$ mapping $\mathbb{R}^{n+m}\rightarrow \mathbb{R}^n$ with invertible jacobian $\frac{\...
7
votes
2
answers
562
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Is the union of a compact and the relatively compact components of its complementary in a manifold compact?
I was thinking of a way to prove this and I realised that for my approach the lemma from the title would be useful, and it´s an interesting question on its own. Obviously it is true if the manifold is ...
- 10.6k
4
votes
0
answers
144
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Examples of indefinite Einstein non-Ricci-flat metric on solvmanifolds
A solvmanifold (resp. nilmanifold) is a compact quotient of a solvable (risp. nilpotent) Lie group $G$. Usually one consider a co-compact lactice $\Gamma$ on $G$ and the solvmanifold is $G/\Gamma$. ...
5
votes
1
answer
206
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Reeb stability counterexample: foliation in $S^{n-2}\times S^1\times S^1$ with non-diffeomorphic leaves
Reeb's global stability theorem requires the foliation to be of codimension 1. As a counterexample, in "Geometric theory of foliations", Camacho and Lins Neto present the following.
Consider the ...
- 135
4
votes
0
answers
131
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Approximation argument in geometric flows
I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken and I'm stuck in the following ...
- 435
5
votes
0
answers
290
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Pullback of Morse form satisfies Palais Smale
Let $(\alpha,g)$ be a Morse-Smale pair on a closed smooth manifold $M$, i.e. $\alpha$ is a Morse form and $g$ a Riemannian metric on $M$ such that stable and unstable manifolds of the gradient vector ...
2
votes
1
answer
720
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Riemannian manifolds: every compact subset is contained in a connected relatively compact open subset [closed]
While working on some problem (not relevant here), it turned out to be convenient to be able to enclose arbitrary compact subsets in "nicer" compact subsets, hence the question:
if $(M,g)$ is a ...
- 5,407
7
votes
1
answer
775
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Fréchet-Kolmogorov compactness Theorem for Lp spaces on manifolds
Suppose I have a family of functions $\mathcal{F} \subseteq L^2(\mathcal{M}, P)$ where $\mathcal{M}$ is a compact manifold, and $P$ is a probability distribution on $\mathcal{M}$. Is there an ...
- 143