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Results tagged with differential-topology
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user 2819
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
4
votes
What is an immersed submanifold?
I think the answer to your final question is no, and more generally: countable unions of embedded submanifolds are precisely the images of (not-necessarily-injective) immersions.
Sketch proof: A co …
- 3,212
2
votes
Submanifolds of $\mathbb{R}^N$ whose local charts have uniformly bounded derivatives
Update: Everything I wrote before is fine, but here's a specific result, from Gilbarg-Trudinger Corollary 16.7 (2001 edition), as an idea of what's possible. Let $\Omega\subseteq \mathbb{R}^n$, and …
- 3,212
9
votes
Derivative of the flow for ODEs on manifolds
As was pointed out by Deane Yang and Igor Khavkine in the comments, this feels like a fact that should be "looser" than Riemannian geometry. Indeed, as I will show below, your formula makes sense in …
- 3,212
0
votes
Is this subset of matrices contractible inside the space of non-conformal matrices?
Yes, they are contractible in $\mathcal{NC}$, in fact even with $SL$ rather than $GL$. First simultaneously rotate $\mathcal{F}$, so the typical element is sent to
$$
\begin{pmatrix} \cos a & -\sin a …
- 3,212
13
votes
1
answer
622
views
Can a PDE constrain the degree of a $C^\infty$ map germ?
Let $\pi:E\to M$ be a smooth vector bundle over a smooth manifold, with $\text{rank}(E)=\text{dim}(M)$. For a section $\sigma$ of $E$ with a zero at $p\in M$, define the degree of the zero at $p$ to …
- 3,212