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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
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Smoothness does not depend on the choice of atlases

Here is a part of a lecture note: I need some help to solve the exercise. I want to show that if $\psi\circ f\circ\phi^{-1}$ is differentiable and $\alpha, \psi$ and $\beta,\phi$ are in the same charts, i.e, if $\psi^{-1}\circ\alpha$ and…
Xena
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Every point in a k-manifold has a neighborhood diffeomorphic to $\Bbb{R}^k$

The problem comes from Alan Pollack's Differential Topology, pg. 5. Suppose that X is a k-dimensional manifold. Show that every point in X has a neighborhood diffeomorphic to all of $\Bbb{R}^k$. I have already shown that $\Bbb{R}^k$ is diffeomorphic…
Pax
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What is the killing field on $S^2$ and $SO[3]$?

What is the killing field on $S^2$ and $SO[3]$? I understand the structure on $S^1$, but not sure about how the vectors work on $S^2$. Thanks in advance!
1LiterTears
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Prove that $\mathcal{M}=\{(x, y) \in X \times Y : f(x) = g(y)\}$ is a smooth submanifold.

Let $\mathcal{X}^m, \mathcal{Y}^n,\mathcal{Z}^p$ be manifolds. Let $f : \mathcal{X} \to \mathcal{Z}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be maps such that $f \pitchfork g$. Prove that $$\mathcal{M}=\{(x, y) \in X \times Y : f(x) = g(y)\}$$ is…
1LiterTears
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$\forall M \in \mathrm{SL}(2, \mathbb{R})$ has a neighborhood diffeomorphic to $\mathbb{R}^3$?

$\forall M \in \mathrm{SL}(2, \mathbb{R})$ has a neighborhood diffeomorphic to $\mathbb{R}^3$? Known that $\mathrm{SL}(2, \mathbb{R})$ is a three-dimensional manifold.
WishingFish
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Does $\mathrm{SL}(2, \mathbb{R})$ have a non-empty boundary?

Does $\mathrm{SL}(2, \mathbb{R})$ have a non-empty boundary? Edit: I've shown that $1$ is a regular value, and hence $\mathrm{SL}(2, \mathbb{R})$ is a three-dimensional manifold as Ted's hint. But I am not sure about what to do next. One more step…
WishingFish
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If $M$ is diffeomorphic to $N$, then $\mathbf{H}_{DR}^p(M)$ is isomorphic to $\mathbf{H}_{DR}^p(N)$.

I thought I got this, but no..... Given $\mathbf{H}_{DR}^2(S^3)$ is trivial but $\mathbf{H}_{DR}^2(T^3)$ is not, how can I show $S^3$ and $T^3$ are not diffeomorphic? I am also wondering about the general case, that if $M$ is diffeomorphic to $N$,…
WishingFish
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What are some applications of differential topology?

I've been studying a lot of differential topology recently, and I like it a lot so far. Out of curiosity, I've tried looking online to see what some applications of differential topology are to things like physics, economics, etc., but most of what…
Nicholas James
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Diffeomorphism of $\mathcal O(n)$ into itself

I have a exercice where I need to proof that $f:\mathcal O(n)\to\mathcal O(n)\;\;,\;\;f(A) =A^{-1}\;$ is a diffeomorphism. $f\;$ is cleary bijective because $\,f\,$ maps a matrix $\,B\,$ to $\,B^{-1}=B^T\,$ since $\,B\,$ is an orthogonal matrix, so…
RAT
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Show that $ (f^*dx_i)(\frac{\partial}{\partial y^j}) = dx_i(f_*(\frac{\partial}{\partial y^j}))$

$U \subset \mathbb{R}^k$ and $V \subset \mathbb{R}^l$ be open subsets. Let $f: V \to U$ to smooth. I am wondering how can I show $$ (f^*dx_i)(\frac{\partial}{\partial y^j}) = dx_i(f_*(\frac{\partial}{\partial y^j}))?$$ I was informed that by…
WishingFish
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Let $M$ be a compact submanifold of dimension $n$ in $\mathbb{R}^{n+1}$ show that exists a normal unitary smooth vector field in $M$

Let $M$ be a compact submanifold of dimension $n$ in $\mathbb{R}^{n+1}$ show that exists a normal unitary smooth vector field in $M$ First i think to these is equivalent to show that $M$ is orientable, then how $\dim(M)=n$ and $M$ is compact then…
Nick_W
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The Generalized Stokes Theorem.

The Generalized Stokes Theorem. If $\omega$ is any smooth $(k-1)$ form on $X$, then $$\int_{\partial X} \omega = \int_X d\omega.$$ Let $C \subset \mathbb{R}^2$ be a (smooth) simple closed curve in the plane and $\Omega$ the closed subset that…
Tumbleweed
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$S^n$ is not diffeomorphic to a product $X \times Y$

Show that $S^n$ can't be decomposed diffeomorphic in a product of manifolds $X \times Y$ with $dim(X), dim(Y) >0$ I try to prove that using tools of differential topology (basically the first 2 chapters of Guillemin Pollack) i think to these way…
Nick
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$(f\circ h)^* \omega = h^*f^*\omega$ - Legit now?

Three pullback identities are required to prove on Guillemin and Pollack's Differential Topology on Page 164. I am not sure if I get it since it is the first time I play with them. I hope I got the first two right, but I was not able to proceed the…
WishingFish
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$f^*(w \wedge \theta) = (f^*w) \wedge (f^* \theta)$

I am reading Guillemin and Pollack's Differential Topology. For the proof on Page 164, I was not able to get through the last step. $$f^*(w \wedge \theta) = (f^*w) \wedge (f^* \theta)$$ Definition. If $f: X \to Y$ is a smooth map and $\omega$ is a…
WishingFish
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