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Results tagged with differential-topology
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user 172802
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”.
40
votes
2
answers
2k
views
Can the nth projective space be covered by n charts?
That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t b …
- 10.6k
14
votes
Accepted
Can a smooth manifold be realised as the image of a smooth function?
My comment turned answer:
Any smooth $m$-manifold $M$ admits a complete Riemannian metric (for example, as this answer says, any manifold embeds into some Euclidean space as a closed subset by Whitney …
- 10.6k
7
votes
2
answers
552
views
Is the union of a compact and the relatively compact components of its complementary in a ma...
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
3
votes
Equidistant points on a compact Riemannian manifold
$K(M,g)$ depends on the metric, as shown by this question, which implies that we can change the metric of $\mathbb{R}^3$ so it has as many points pairwise at distance $1$ as we want.
- 10.6k