For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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1
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1answer
33 views

Prove that if $f$ is a solution of the heat equation.

Let $M^{k}\subset\mathbb{R}^{n}$ be a compact, oriented manifold, and assume that $f:M^{k}\times[0,\infty)\to\mathbb{R}$ is smooth. The heat equation is $$\triangle_{x}f(x,t)=\dfrac{\partial ...
5
votes
3answers
91 views

One-sided submanifolds in Hempel's 3-Manifolds

Early on in Hempel's book 3-Manifolds, he discusses two-sided submanifolds: if $N$ is a manifold of dimension $n$, and $M$ is a submanifold of dimension $(n-1)$, then $M$ is two-sided if there is an ...
0
votes
0answers
50 views

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 ...
1
vote
0answers
21 views

Map of constant rank

Let $f_1, \dots, f_m \colon M \to \mathbb{R}$ be smooth functions on a manifold $M$, such that $F := (f_1, \dots , f_m) \colon M \to \mathbb{R}^m$ has constant rank $k <m $ on M. I'm trying to ...
2
votes
1answer
49 views

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 ...
0
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0answers
14 views

Algorithms operating on manifolds , SO(3)

I am trying to understand the manifold toolkit and its related paper. The paper is nicely explained, but being noobish, I couldn't get an intuitive understanding of it. I understood that it tries to ...
4
votes
1answer
49 views

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 = ...
3
votes
2answers
105 views

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 $d\alpha = \omega$?
3
votes
0answers
43 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 ...
0
votes
1answer
39 views

What is the manifold underlying the Lie group $SU(p,q)$?

I've been trying to google around this topic without success, apologies in advance if I missed an obvious resource. I'm trying to understand what manifold (compact or not) underlies the complex Lie ...
2
votes
1answer
62 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 $\int_A \alpha \in \mathbb{Z}$, ...
3
votes
0answers
59 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 ...
2
votes
0answers
65 views

Visualizing the evolution of a Riemannian metric

I'm doing some reading into Riemannian geometry and PDEs and I have the following question. When we evolve a Riemannian metric (by say the Ricci flow) we are evolving a bilinear form on a manifold ...
4
votes
1answer
48 views

Two ways of inducing a metric/topology on a manifold.

Consider $\Bbb R^3$ equipped with the usual euclidean metric and topology. And consider the subset $S^2 := \{ x\in\Bbb R^3\,|\,d(x,0)=1\}\subset\Bbb R^3$ . Suppose we wanted to make $S^2$ in to a ...
0
votes
1answer
37 views

Continuous maps on smoth manifolds

Let $M$ be a smooth manifold, $f:M \to \mathbb{R}$ be a $C^{\infty}$ map and $f(p)=0$. **My question:**Does there exist a neighborhood $U$ of $p$ in $M$ such that $f(U)=0$? i know by coordinate ...
1
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0answers
41 views

Embed 3-manifold being homology sphere to $S^4$

Can someone explain for me following sentence "Mike Freedman has proven all homology 3-spheres admit tame topological embeddings into $S^4$.", which I found on "open problem garden" ...
1
vote
1answer
17 views

Guaranteed smooth surfaces

The sets $X_a$ and $Y_b$ of equations $x^2+y^3+z = a$ and $x+y+z = b$, respectively, are smooth surfaces in $R^3$. What values of $a$ and $b$ guarantee that the intersection of $X_a$ and $Y_b$ is a ...
1
vote
1answer
26 views

Isotopies with fixed subsets

Let $M$ be a smooth manifold. Let $f:M\rightarrow M$ be a diffeomorphism which is smoothly isotopic to the identity. Let $X\subset M$ be a compact subset such that $f|_X = id_X$. Under what ...
0
votes
1answer
10 views

arcs transversal to $C^k$ maps

In reference to: "if $M \subset\mathbb{R}^2$ and $f\in C^k(M,\mathbb{R}^2)$, then for each regular $x\in M$, we can find an arc $\Sigma$ containing x which is transversal to f" what is meant by an ...
0
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1answer
27 views

Is it possible to define a differentiable manifold structure on a cone?

A cone is a topologic manofold but can we define a differentiable manifold structure on it?
0
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0answers
40 views

Is this a compact manifold?

Consider $X=\left\{0,1,2\right\}^{\mathbb{Z}}$. My very short, and hopefully not too stupid question is, if $X$ is a compact manifold. I think compact is clear by Tychonoff's theorem, but I do not ...
2
votes
1answer
42 views

Description of Pfaffian System in Chern's Lectures on Differential Geometry

Here is a quotation from pg. 81, section 3-2 of Chern's Lectures on Differential Geometry: Suppose $L^r=\{X_1,\ldots,X_r\}$ is a smooth $r$-dimensional distribution on $M$. [$M$ is an ...
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 ...
1
vote
0answers
18 views

Composition of analytic functions is analytic in Manifolds

My problem is in analytic manifolds.According to Cohn's book a function $f$ in a manifold $M$ is analytic at $p \in M$ if it can be expressed as a power series of $\sigma(p)=(x_{0})$. That means ...
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
159 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
43 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.) ...
3
votes
0answers
36 views

Riemannian surface, identity relating scalar curvatures and Laplacian [closed]

Let $S$ be a Riemannian surface, i.e. a $2$-dimensional manifold, with metric $g$. Define a new metric $\tilde{g}$ by $\tilde{g} = e^fg$ for some smooth function $f$. If $s_{\tilde{g}}$ and $s_g$ are ...
2
votes
1answer
67 views

Is a Kähler manifold necessarily symplectic?

Let $M$ be a Riemannian manifold. If we pick a basepoint $p \in M$, then for any smooth path $\gamma: [0, 1] \to M$, parallel transport along $\gamma$ induces an automorphism $g_\gamma \in ...
0
votes
1answer
31 views

How do I prove that a surface defined by two functions is a manifold?

I am trying to show that $$\{(x, y, z)\mid z=x^2+y^2, z\leq 2\}$$ is a manifold. I am trying to express it as a set where $f(x, y, z)\geq 0$ for some smooth $f$ on an open set, but as the set ...
3
votes
1answer
60 views

Riemannian manifold, $\alpha \in \Omega^p(M)$ parallel implies $\alpha$ is closed?

Let $M$ be a Riemannian manifold, and let $\alpha \in \Omega^p(M)$ be parallel; i.e. suppose $\nabla \alpha = 0$ where $\nabla$ is the Levi-Civita connection. Does it necessarily follow that $\alpha$ ...
1
vote
1answer
26 views

Intuition of section of a hermitian line bundle

Can someone explain to me intuitively and without much technical stuff the following: A hermitian line bundle is a complex line bundle with a hermitian metric. I think of this as a bundle over my ...
2
votes
1answer
29 views

Compact 3-manifold implies finite triangulability

I know that it's a theorem by Moise that every compact 3-manifold admits a finite triangulation but to me the astounding part of that statement is the existence part instead of the finite one. So I ...
1
vote
2answers
44 views

Smooth functions between manifolds and subsets of manifolds

I'll be quoting from the Wikipedia page on smoothness. Smooth function between manifolds are defined as follows: If $F$ is a map from an $m$-manifold $M$ to an $n$-manifold $N$, then $F$ is smooth ...
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 + ...
0
votes
1answer
16 views

Continuity of the Length of shortest arc containing angles

Here's an interesting question! Let $x=(x_1,x_2,\dots,x_n)$ points on the unit circle of $\mathbb{R}^2$. And let $V_{\min}(x)$ be the length of the shortest arc containing all the points ...
0
votes
1answer
34 views

How do I prove that a set of equations define a submanifold?

I have two equations of 4 variables, and want to prove that they define a submanifold. I showed that two of the variables are defined by the other two, and wanted to get a graph out of it to show ...
1
vote
2answers
48 views

Understanding of tangent space

I have some understanding issues with the following theorem. Let $M \subset \mathbb{R}^N$ be a d-dimensional smooth manifold. $p \in M$ with $(\phi, U)$ a chart around $p$. Then $T_pM = (d\phi ...
0
votes
1answer
34 views

Smooth function on a manifold not dependent on coordinate chart

I'm having trouble with the proof of the following remark from page 59 of Tu's book on Manifolds. The part I'm worried about is where he gets that $\phi\circ \psi^{-1}$ is $C^\infty$. Is he allowed to ...
0
votes
0answers
16 views

Proving that the following is a submanifold with boundary

Let $\mathbb{H}^{k} = \{ \ \mathbb{x} = (x_{1}, ..., x_{k}) \in \mathbb{R}^{k}\ | \ x_{k} \geq 0\ \}$. Definition: If the following holds, then $\mathcal{M}$ is a $k$-submanifold of $\mathbb{R}^{n}$ ...
1
vote
1answer
28 views

Mapping the Upper Half plane to the Unit Ball

Consider the sets $\mathbb{H} = \{ \ (x_{1}, ..., x_{n}) \in \mathbb{R}^{n} \ | \ x_{n} \geq 0\ \}$ and $B_{1}(\mathbf{0}) = \{ \ \mathbf{x} \in \mathbb{R}^{n}\ | \ ||\mathbf{x}|| \leq 1 \ \}$. Does ...
1
vote
1answer
29 views

Examples of quotient manifolds which are not locally trivial fibrations?

Let $X$, $Y$ be differentiable manifolds, and $f : X \to Y$ a smooth surjection. Then $Y$ is said to be a quotient of $X$ if 1) $Y$ has the quotient topology 2) A function $g : Y \to \mathbb{R}$ is ...
0
votes
2answers
22 views

Proving non-compactness of a manifold

I have been trying to solve the following problem: Let $M \subset \mathbb R^3$ be the set of points $(x,y,z) \in \mathbb R^3$ at which $xy + xz + yz = 1.$ Prove that $M$ is a $2$-dimensional manifold. ...