Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [differential-topology]

Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

7287 questions
4
votes
1 answer

A submanifold of a submanifold is a submanifold is the ambient space.

Definition: Let $X$ be a manifold and $Y \subset X$. We say that $Y$ is a submanifold of $X$ if for every $y \in Y$, there exists some chart $(U, \phi)$ and $p \leq n$ such that $\phi(U \cap Y) = \phi(U) \cap (\mathbb{R}^p \times \{0\}^{n - p})$…
Hermès
  • 2,647
4
votes
1 answer

When is the topological closure of a submanifold a submanifold with boundary?

Let $M$ be a smooth manifold and $N\subset M$ a connected embedded submanifold which is not topologically closed. Assume that $\bar{N}\setminus N$ is a smooth embedded submanifold (where $\bar{N}$ is the closure of $N$). Question: Then is $\bar{N}$…
Matthew Kvalheim
  • 1,990
4
votes
1 answer

Question about nested manifolds with boundary which are diffeomorphic

Suppose $M, N$ are compact manifolds with boundary, with $M \subset \text{int}(N)$, and assume that $M$ is diffeomorphic to $N$. Is $N\setminus M$ automatically diffeomorphic to $\partial N \times [0,1]$? If not, what is a counterexample? I suspect…
Matthew Kvalheim
  • 1,990
4
votes
1 answer

Example of symplectic manifold

I wonder why tangent bundle is not symplectic. As you know, cotangent bundle is symplectic. (1) question 1 : Is cotangent bundle isometric to tangent bundle ? (2) question 2 : Why is not tangent bundle symplectic manifold ?
HK Lee
  • 19,964
4
votes
0 answers

Embed connected sum of submanifolds in the ambient manifold

The book Differential Manifolds by Kosinski claimed we can embed the connected sum of two submanifolds of codimension $>1$ into the ambient manifold. I am wondering why we need the extra condition $>1$ at here. At least in dimension 1-3 I could not…
Bombyx mori
  • 19,638
  • 6
  • 52
  • 112
4
votes
1 answer

Continuity of the figure-eight map

Let $\beta(t) = (\sin 2t, \sin t) :(-\pi, \pi) \to \mathbb{R}^2$. This is an injective immersion and so its image $S$ is an immersed submanifold of $\mathbb{R}^2$ (the figure-eight). Let $G(t) = (\sin 2t, \sin t) : \mathbb{R} \to S$ where $S$ is…
cleone
  • 449
4
votes
2 answers

Smooth embedding that isn't an open or closed map

In Lee's book there is an exercise: Give an example of a smooth embedding that is neither an open map nor a closed map. I'm confused; a smooth embedding is a homeomorphism which is both an open and closed map. Am I missing something here?
Jonathan Gafar
  • 1,091
4
votes
0 answers

Do we really need embedding for this?

Question 8-8 in Lee's Introduction to smooth manifolds asks us to show that if $M \subset N$ is an embedded submanifold then it is closed iff the inclusion map is proper. Equivalently, a smooth embedding $g:M \to N$ is proper iff its image is…
Bernard
  • 927
4
votes
0 answers

Baby Sard's theorem

Does there exists a "simple" proof that every smooth map $f: M\to N$ between manifolds contains at least one regular value, without using Sard's theorem? Motivation: In Milnor, Brouwer's fixed point theorem is proved by means of Sard's theorem, but…
Peter Franek
  • 11,522
4
votes
1 answer

Embedded submanifold is closed in some open set of total manifold

Suppose $M$ is a differential manifold, $S$ an embedded submanifold, prove $S$ is closed in some open set of $M$. And I think it still holds up when $S$ is an immersed submanifold.
user360777
  • 402
4
votes
1 answer

The inclusion map is not an embedding?

In 4b it states: Show that the inclusion map is an embedding. The solution say the question is faulty and proceeds to say Using the definition of embedding given by Guillemin and Pollack the inclusion may not be an embedding. In which case the…
Dair
  • 3,064
4
votes
1 answer

Showing a certain form is exact

I'm trying to solve the following: Let $f: S^{2n - 1} \rightarrow S^n$ be a smooth map, and let $\omega$ be an n-form on $S^n$ such that $\int_{S^n} \omega = 1$. Show that $f^*\omega$ is exact, and if $f^*\omega = d\alpha$, then $\int_{S^{2n-1}}…
Joe
  • 1,310
4
votes
0 answers

Let $M$ be a manifold noncompact. Then there is a closed embebdding of the half line $[0, \infty)$ into $M$.

Let $M$ be a manifold connected hausdorf noncompact. Then there is a closed embebdding of the half line $[0, \infty)$ into $M$. I'm hard to build such a function without self-intersections. Book: Differential Topology, Hirsch. p 27.
Henfe
  • 1,208
4
votes
0 answers

Given two points on a manifold, is there a chart containing both?

Consider a connected manifold $M$ and two distinct points $p,q$ on the manifold. Is it true that there exists a chart containing both? How can one prove this result? OBS: I've seen an answer on the internet which relied on taking a path from $p$ to…
Aloizio Macedo
  • 34,292
4
votes
1 answer

Let $\Omega$ be a star-shaped open set of $\mathbb{R}^3$. Under which conditions is $\Omega$ analytically diffeomorphic to $\mathbb{R}^3$?

In this post A star shaped open set in $\mathbb{R}^n$ is diffeomorphic to $\mathbb{R}^n$ I found a proof that $\Omega$ is always diffeomorphic to $\mathbb{R}^3$. In which cases can such a diffeomorphism be analytical?
slatobardi
  • 41
1 2 3
…
40 41