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.

learn more… | top users | synonyms

1
vote
1answer
35 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 ...
2
votes
1answer
42 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 ...
1
vote
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 ...
0
votes
1answer
35 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 ...
4
votes
1answer
47 views

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 ...
1
vote
0answers
75 views

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 ...
3
votes
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 ...
6
votes
1answer
56 views

$M$ 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 ...
2
votes
1answer
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
votes
1answer
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 ...
2
votes
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 ...
6
votes
3answers
158 views

$2$-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 ...
3
votes
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 ...
5
votes
1answer
52 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
votes
1answer
60 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
votes
1answer
42 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 ...
10
votes
1answer
111 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}} ...
7
votes
1answer
52 views

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.) ...
0
votes
0answers
28 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 ...
3
votes
0answers
36 views

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 + ...
1
vote
1answer
44 views

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
votes
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 ...
3
votes
1answer
58 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 ...
1
vote
1answer
33 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 ...
3
votes
1answer
42 views

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 ...
0
votes
0answers
35 views

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) $ ...
2
votes
1answer
49 views

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 ...
2
votes
1answer
38 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?
2
votes
1answer
99 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 $U_i$ ...
12
votes
2answers
141 views

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 ...
1
vote
0answers
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: ...
0
votes
0answers
26 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
votes
1answer
26 views

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
votes
1answer
61 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 $k$ ...
0
votes
0answers
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 ...
0
votes
0answers
29 views

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 ...
5
votes
5answers
532 views

A Compact Hausdorff Space with no Manifold Structure? [closed]

What is an example of a compact Hausdorff space that cannot be given the structure of a (i) differential manifold (ii) topological manifold?
0
votes
0answers
40 views

Topology for distributions on a compact space

I'm having trouble in distribution theory, though not in the usual setting. The context is in theoretical physics, trying to solve BF theory. My goal is to solve the following equation: $$\forall i ...
5
votes
1answer
170 views

Topology of Isolated Singularities

In Singular Points of Complex Hypersurfaces, Milnor shows that if $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$, holomorphic, has an isolated singularity at the origin, then $f^{-1}(\epsilon) \cap ...
3
votes
1answer
78 views

Property of second Steifel-Whitney class?

Let $M$ be manifold, $n = 4$. Is $w_2$ special in in the regard it's the only thing of $H^2(M, \mathbb{Z}_2)$ where $w_2 \cup \tau = \tau \cup \tau$, $\tau \in H^2(M, \mathbb{Z}_2)$ or not? I wondered ...
0
votes
1answer
15 views

Regarding embeddings and homological/cohomological injectivity

Possibly low-level question here: if one has an embedding $M\hookrightarrow X$, call it $\iota$, then this is of course an immersion, so the differential maps $\iota_{\ast}:T_{p}M\rightarrow ...
2
votes
0answers
90 views

Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
8
votes
6answers
109 views

Examples of global properties that don't arise from local knowledge

Let $M$ be a smooth manifold. As an example of a global property that arises from local data we know that if $(M,g)$ is a compact surface without boundary then the Euler characteristic is given by $$ ...
5
votes
0answers
78 views

Linking circles inside an immersed surface

A smooth embedding $f : D \to \mathbb{R}^3$ can be isotoped to a canonical inclusion $D \hookrightarrow \mathbb{R}^3$. (This is part of a proof that only the unknot has the disk as a Seifert surface.) ...
7
votes
1answer
56 views

Is there any result on the “counting” of minimal atlas?

Take a differentiable manifold $M$. Define $\eta(M)$ as $\min\{\#\mathfrak{A} \mid \mathfrak{A} \text{ is an atlas for $M$}\}$. For example, if $M=S^n$, we have that $\eta(M)=2$, since $S^n$ is ...
2
votes
0answers
37 views

singular $1$ cube - Boundary of $2$ chain

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
0
votes
0answers
25 views

Independence of parametrization in defining integral of differential form

This is an exercise from Spivak's Calculus on Manifolds. Questions asks the following : (Independence of parametrization). Let $c$ be a singular $k$ cube and $p:[0,1]^k\rightarrow [0,1]^k$ be ...
6
votes
1answer
69 views

Is $[N]^\#([N])$ congruent to $w_n(\nu_N)([N])$ mod $2$, where $\nu_N$ is the normal bundle of the embedding of $N$ in $M$?

Let $M$ be a closed, smooth, orientable $2n$-manifold, and let $N$ be a closed, smooth, orientable $n$-submanifold. Let $[N]^\#$ denote the cohomology class (Poincaré) dual to the homology class ...
0
votes
0answers
74 views

When is a diffeomorphism analytic?

I've read somewhere "a class $C^\infty$ diffeomorphism is said to be analytic" but I forgot to write down where I read this and now I can not find it, which makes me wonder if it's true? I'm working ...
2
votes
1answer
58 views

Why is even codimension necessary to apply excision for the Euler characteristic?

In this answer on MathOverflow, it is claimed that $$\chi(X/Z)=\chi(X)-\chi(Z)$$ holds for complex subvarieties $Z$ only because $Z$ has even codimension. It is implied that for $Z$ with odd ...