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Results tagged with smooth-manifolds
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user 172802
Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].
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
Accepted
Do non-zero derivatives imply tangent lines (and vice versa)?
The answer to question $1$ is yes: we can suppose $t=0,\gamma(0)=(0,0)$ and $\gamma'(0)=(1,0)$ for the purposes of this question. Then as $\frac{||\gamma(x)||}{|x|}\to 1$ when $x\to0$, there is some $ …
- 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