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Results tagged with differential-topology
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user 485792
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”.
8
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1
answer
1k
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Philosophy behind the Ricci flow
I don't know if my question is too simple for this forum but let me proceed.
In Ricci flow one equips a smooth manifold $M$ with a Riemannian metric $g_0$ and evolves the metric with "time": giving ri …
- 521
-2
votes
1
answer
188
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Topologies in the vicinity of Euclidean space
Given a smooth function $f:\mathbf R^n\to \mathbf R^m$ with $0$ as a regular value, I define the $(n-m)$ dimensional smooth manifold $M_f:=f^{-1}(0)$.
Let $f_0(x_1,...,x_n):=(x_1,...,x_m)$; $M_{f_0}$ …
- 521
6
votes
1
answer
937
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Can a smooth manifold be realised as the image of a smooth function?
Consider, $M$, a smooth $m$ dimensional submanifold of $\mathbf R^n$. Does there exist a smooth map $X: \mathbf{R}^m\to\mathbf R^n$ such that $M=X(\mathbf R^m)$?
$X$ may have points at which the Jacob …
- 521
2
votes
2
answers
380
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Under what conditions can an orientable Riemannian 3-manifold be defined implicitly?
Under what conditions can an orientable Riemannian 3-manifold $\Sigma$ be defined implicitly?
What I mean by implicitly is that there exists a smooth function $f:\mathbb{R}^n\to \mathbb{R}^m$, such th …
- 521