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.

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Topological manifold but not “smooth”

I would like to know if there are examples of topological manifolds that not admit a smooth atlas, or more generally a differentiable structure of class $C^k$ for $k\geq1$. In the book "Introduction ...
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Is $H([f]) = \int_{S^{2n - 1}} \alpha \wedge d\alpha$ independent of all choices, defines a map $H: \pi_{2n - 1}(S^n) \to \mathbb{Z}$?

Let $[f] \in \pi_{2n - 1}(S^n)$. Choose a smooth representative $f: S^{2n - 1} \to S^n$. Let $\omega$ be a smooth $n$-form on $S^n$ with$$\int_{S^n} \omega = 1.$$We know that$$f^*\omega = d\alpha$$for ...
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Smooth representative $f: S^{2n - 1} \to S^n$, do we have $f^*\omega = d\alpha$?

Let $[f] \in \pi_{2n - 1}(S^n)$. Choose a smooth representative $f: S^{2n - 1} \to S^n$. Let $\omega$ be a smooth $n$-form on $S^n$ with$$\int_{S^n} \omega = 1.$$Do we have that$$f^*\omega = ...
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21 views

A disk cross sphere in a sphere

Let $\lambda$ and $\mu$ be positive integers and define $D(\lambda, \mu) = \{ p \in S^{\lambda + \mu -1} : \sum_{i \leq \lambda} x_i^2 \geq \sum_{i > \lambda} x_i^2 \}$. Why is $D(\lambda, \mu)$ a ...
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Two neighborhoods of $0$ in the plane and the upper half plane (resp.) cannot be diffeomorphic. [duplicate]

Let $\mathbb{R^2}$ be equipped with the standard topology, and let $\mathbb{H^2}$ be the upper half plane (containing the x-axis), equipped with the subspace topology. Let $U$ be an open neighborhood ...
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Does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?

Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$. As the question title suggests, does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?
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45 views

Two smoothly homotopic smooth maps induce same maps on de Rham cohomology

Let $a$, $b: M \to N$ be smoothly homotopic smooth maps. How do I see directly that $a$ and $b$ induce the same maps on de Rham cohomology? I know I want to construct a suitable chain homotopy between ...
3
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2-connected 6 manifolds with boundary $S^5$

What are the 2-connected 6-manifolds that have boundary $S^5$? Are they all of the form $(\sharp_{i=1}^k S^3 \times S^3) \backslash D^6$ for some $k \ge 1$? Also, I think if $M^5$ is ...
2
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1answer
63 views

Wedge product of closed form each with integral periods has integral period?

Suppose $\alpha$ and $\beta$ are closed forms on $M$ which have integral periods, i.e. for all $[A] \in H_*(M, \mathbb{Z})$ represented by a smooth cycle $A$, we have $\int_A \alpha \in \mathbb{Z}$, ...
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60 views

how to prove that every low-dimensional topological manifold has a unique smooth structure up to diffeomorphism?

It is a deep fact of low-dimensional manifold topology that the notion of isomorphism coincides for the three categories Top, PL, and Diff. In other words, every topological manifold of dimension ...
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Can any path in the diffeomorphism group of a smooth manifold be approximated by a smooth one?

Given a smooth compact $n$-dimensional manifold $M^{n}$, let $\operatorname{Diff}(M)$ denote the group of smooth diffeomorphisms $M \rightarrow M$ equipped with the Whitney $C^{\infty}$-topology. Let ...
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31 views

How are related Connected Component with Open-Closed Subset

I used to think that connected components were closed and open at the same time, but I discovered quite recently (italian page of Wikipedia) that this might not be the case. What I'm asking for is: ...
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21 views

Diffeomorphisms with fixed subsets and isotopies

I asked a much more general version of this question previously, but I realize now that it was so broad that there probably are no good general answers, so I want to make a more specific question. ...
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1answer
37 views

Degree of infinite dimensinal antipodal map

Let $E$ is a infinite dimensional Banach space and $\Omega=\{x\in E: ||x||=1\}$ . $L:\Omega\rightarrow\Omega,L(x)=-x$, how to compute the degree of $L$? In fact ,I just know that the algebra define ...
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1answer
42 views

Representation of linearly independent and commuting vector fields with local coordinate basis of the tangent space

I am a beginner of differential geometry. I wonder if the following proposition is true: Let $M$ be an n-dimensional manifold and $X_1, \dots ,X_m(m \le n)$ be m commuting and linearly ...
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1answer
45 views

Stronger Whitney approximation for smooth manifolds

If $M$ and $N$ are smooth manifolds, then any continuous map $f:M\rightarrow N$ is homotopic to a smooth map $g:M\rightarrow N$. If $f$ is smooth on a closed set $K\subseteq M$, the homotopy can be ...
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1answer
24 views

Extending differentiable functions to the whole manifold

Let $M$ be a differentiable manifold, $U\subseteq M$ an open neighborhood of $p\in M$ and $f:U\to\mathbb{R}$ differentiable. Then there exists $F:M\to\mathbb{R}$ differentiable such that $F=f$ in a ...
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1answer
37 views

Proving that the tangent space of $\Delta$ is the diagonal of $T_pM\times T_pM$

Let $M$ be a differentiable manifold. Prove that the tangent space of $\Delta=\{(p,p):p\in M\}\subseteq M\times M$ at a point $(p,p)$ is the diagonal of $T_pM\times T_pM$. There are many things ...
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How can I draw plane distributions in $\mathbb{R}^3$?

I see so many nice pictures of contact structures, integrable plane distributions, etc., in manuscripts and online and I have absolutely zero idea how they're made. For example, the following image ...
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Understanding Higher Orders and Levels of Mathematics [closed]

this is a more general question than the most. As a background, I'm currently a graduate student getting my masters in mathematics. I have what I believe is a strong passion for mathematics, though ...
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1answer
33 views

Proving that $\mathbb{R}P^n$ is a manifold

Consider $\mathbb{R}P^n$ as the quotient space of $S^n$ with antipodal points identified. Prove that $\mathbb{R}P^n$ is a manifold of dimension $n$. (I'd like to clarify that I've seen the ...
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$M$ closed $3$-manifold, $\xi$ integrable $2$-dimensional subbundle of $TM$, ensuing properties.

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. There is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a ...
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47 views

Poisson bracket makes $C^\infty(M)$ into a Lie algebra

Let $M$ be a symplectic manifold with symplectic form $\omega$. Define the Poisson bracket of two smooth functions $f$, $g$ by $\{f, g\} := \omega(X_f, X_g)$. How do I see that $X_{\{f, g\}} = [X_f, ...
4
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42 views

Symplectic manifold, flow $\phi_t$ generated by unique vector fiel $X_f$ preserves symplectic form $\omega$?

Let $M$ be a symplectic manifold with symplectic form $\omega$, let $f$ be a smooth function on $M$, and let $X_f$ be the unique vector field on $M$ so that $df(Y) = \omega(X_f, Y)$ for all vector ...
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2answers
51 views

Symplectic manifold $M$, unique vector field $X_f$ on $M$ so that $df(Y)= \omega(X_f, Y)$ for all vector fields $Y$?

Let $M$ be a manifold. A symplectic form is a closed $2$-form $\omega$ suh that $X(p) \to \omega(X,\cdot)(p)$ is an isomorphism from $T_pM$ to $T_p^*M$ for each $p \in M$. A manifold together with a ...
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$2$-dimensional subbundle of tangent bundle of closed $3$-manifold integrable if and only if $\alpha \wedge d\alpha = 0$?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. From here and here, I know that there is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any ...
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1answer
30 views

Open immersion pulls back symplectic form to symplectic form?

If $M$ is symplectic, and $f: W \to M$ is an open immersion, i.e. an immersion where $W$ and $M$ have the same dimension, does $f$ necessarily pull back a symplectic form on $M$ to a symplectic form ...
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53 views

Any two $1$-forms $\alpha$, $\alpha'$ with property satisfy $\alpha = f\alpha'$ for some smooth nowhere zero function $f$?

This is a followup question to here. Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. Is there a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ ...
4
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1answer
61 views

Is there a nowhere $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a section of $\xi$?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. Is there a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a ...
7
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1answer
44 views

If and only if criterion for something to be a differential ideal

Let $I \subset \Omega^*(M)$ be a ($2$-sided) ideal (i.e. $I$ is a vector subspace, and for any $\alpha \in I$ and $\omega \in \Omega^*(M)$ we have $\omega \wedge \alpha \in I$). We say $I$ is a ...
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1answer
114 views

Flows and Lie brackets, $\beta$ not a priori smooth at $t = 0$

Let $X$ and $Y$ be smooth vector fields on $M$ generating flows $\phi_t$ and $\psi_t$ respectively. For $p \in M$ define$$\beta(t) := \psi_{-\sqrt{t}} \phi_{-\sqrt{t}} \psi_{\sqrt{t}} ...
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1answer
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Reference on manifolds with corners

Is there a systematic treatment of (finite dimensional) manifolds with corners in the literature which carefully introduces all usual differential topological notions (submanifolds, embeddings, etc.) ...
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29 views

How to show this atlas is maximal on the sphere $S^n$?

I have the following problem: Let $S^n=\{x\in\mathbb{R}^{n+1}:\|x\|=1\}$ and $\pi_{\pm}:S^n\setminus\{{p_{\pm}}\}\to\mathbb{R}^n$ be the stereographic projections from the poles ...
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Primary obstruction to the existence of a cross-section of $V_{n - q}(\omega)$ is a cohomology class in $H^{2q+2}(B, \pi_{2q+1} V_{n - q}(F))$?

Let $V_{n - q}(\mathbb{C}^n)$ denote the complex Stiefel manifold consisting of all complex $(n - q)$-frames in $\mathbb{C}^n$, where $0 \le q < n$. This manifold is $2q$-connected, and$$\pi_{2q + ...
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Fundamental group - space of copies of circle $S_1$

For $n>1$ an integer, let $W_n$ be the space formed by taking $n$ copies of the circle $S_1$ and identifying all the $n$ base points to form a new base point, called $w_0$. What is $\pi_{1}$($W_n , ...
4
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1answer
34 views

Extending the definition of partial differention to nonopen domains

Most of the time , When we talk about the partial derivatives of a function of several varibles, we need to require the function'domain is an open subset. But under some special cases, the ...
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1answer
60 views

On the $\omega$-limit set of a trajectory converging to a submanifold

Let $x\in M$, where $M$ is an $m$-dimensional smooth manifold. Let $N$ be an $n$-dimensional smooth and compact manifold so that $n<m$. Let $X:M\to TM$ be a smooth vector field and denote by ...
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1answer
39 views

The hopf bundle over $S^1$ is a trivial bundle.

The hopf bundle over $S^1$ is the bundle obtained after two twists. I was wondering if this bundle is the trivial bundle. Intuitively it seems like it should not be trivial since there are two twists ...
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1answer
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Extension of Smooth Functions on Embedded Submanifolds

In Lee Smooth Manifolds, this problem is given: if $S \subset M$ is smoothly embedded and every $f \in \mathcal{C}^{\infty}(S)$ extends to a smooth functional on $\textit{all}$ of $M$, then $S$ is ...
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Finding the Euler parametrization of a curve

I have the following question as a homework problem for my differential geometry class: find the curvature and the explicit Euler parametrization of the ellipse $ \gamma(t) = (a \cos t, b \sin t) $ ...
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1answer
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Every point of Ext(C) lies on some tangent line to C

In our differential geometry class our professor gave the following problem as a homework problem: Let $ C $ be a smooth non-singular simple closed curve in plane. Prove that every point of the ...
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1answer
39 views

Can we construct a continuous but nowhere differentiable surface?

In my analysis course we learned about the Weierstrass function which is continuous but nowhere differentiable, is it possible to make a surface which is continuous and nowhere differentiable?
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1answer
103 views

Show that the Mobius strip is non-orientable

The Mobius strip is the 2D manifold $M$ with the atlas of $n$ cubic charts $U_i$, $1 ≤ i ≤ n$, with coordinates $(x_i, y_i)$ satisfying $|x_i| < 1, |y_i| < 1$. Let $U_i^±$ be a part of $U_i$ ...
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How do we construct an associated bundle $V_{n, q}(\omega)$ over $B$ with typical fiber $V_{n - q}(F)$?

Let $V_{n - q}(\mathbb{C}^n)$ denote the complex Stiefel manifold consisting of all complex $(n - q)$-frames in $\mathbb{C}^n$, where $0 \le q < n$. This manifold is $2q$-connected, and$$\pi_{2q ...
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76 views

Advanced definition of derivative.

In the paper On proof and progress in mathematics, W. P. Thurston gives the following interpretation of the derivative. ...one person’s clear mental image is another person’s intimidation: ...
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27 views

Checking if a tangent bundle is trivial

I don't know how to check if tangent bundle $TP$ of the surface $P$ is trivial. Are there any general methods to deal with this problem? For example how to check it for $P\subset \Bbb{R}^3$ arisen by ...
2
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1answer
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Discrete subgroup of Lie group has properly discontinuous action

I've found some literature which would be helpful if I understood the following, "we can choose neighborhood $U,V$ of the identity such that $VV^{-1} \subset U$ and $U \cap \Gamma = \{e\}$. " ...
5
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1answer
63 views

Where is the error in this proof of the Hodge theorem?

Let $(M,g)$ be a closed smooth Riemannian manifold. The following is the decomposition part of the Hodge theorem: Theorem The canonical map $\mathscr{H}^k(M)\to H^k(M)$ from harmonic $k$ ...
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21 views

Plane integral for continuous curves

I'm trying to understand complex path integral $\int_C f(z)dz$ for continuous closed curve $C$. Is it necessary that $C$ is rectifiable and not just generally continuous? Do we get all the ...
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Partitions of unity subordinate to open cover of manifolds with boundary?

I am attempting to adapt Lee's proof of the fact that open covers of manifolds without boundary always admit smooth partitions of unity to the case in which the manifold does have boundary. The ...