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

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46 views

Why don't I get the same answer when I calculate the pullback vs integral over a manifold?

Let's take the differential form $\omega = xy \, dx \wedge dy$. We say that $M$ is the surface $z = x^2 + y^2 \leq 1$ with the standard orientation. I can calculate $\int_M \omega$ via pullback and ...
2
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0answers
33 views

Extension theorem from Guillemin-Pollack, motivated sketch of proof?

Let $W$ be a compact, connected, oriented $k + 1$ dimensional manifold with boundary, and let $f: \partial W \to S^k$ be a smooth map. Could anybody sketch with good motivation that $f$ extends to a ...
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1answer
32 views

Smoothness of projective hypersurface

I'm trying to understand the question and answer here, but I don't quite follow what they're doing, so here is my take on it. The problem is to show that in $\mathbb R P^2$, given a homogeneous ...
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1answer
45 views

Fundamental theorem on flows lee's book 2nd edition

I am reading Lee's book Introduction to smooth manifolds 2nd edition chapter 9 the fundamental theorem on flows. In the proof of the fundamental theorem on flows the author defines ...
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1answer
38 views

Notation (manifolds, harmonic analysis)

I was reading the paper "On the Multilinear Restriction and Kakeya conjectures" by Bennett, Carbery and Tao For each $1\leq j\leq n$ let $U_j$ be a compact neighborhood of the origin in ...
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1answer
46 views

Does $X^T Y=I$ define a manifold?

Consider the set of pairs (X,Y) of real $n\times k$ matrices ($k\le n $) defined by $X^T Y=I$. Is this set a manifold? And if so, what is the tangent space at a point $(X_0,Y_0)$?
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0answers
57 views

does there exist a generalization of manifold

is there a generalization of a manifold, where instead of being locally like $\mathbb{R}^n$, it is locally some other space?
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0answers
31 views

Flow of Normalized Gradient Field of a Smooth Function

Suppose $f: \mathbb R^n \rightarrow \mathbb R$ is a smooth function, with a finite number of critical points (which are then isolated). Let us take as a vector field on $\mathbb R^n$ (minus those ...
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1answer
24 views

Basis for the tangent space $T_t[0, 1]$?

I guess this question has more to do with notations then with concepts. Let $I:=[0, 1]$ and $\alpha:I\longrightarrow M$ a path with values in a smooth manifolds $M$. I'm reading a paper in which the ...
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0answers
32 views

Calculating 3-dimensional Volume of 4-dimensional Graph

Let $D = \{(x,y,z) \in R^3 | |x|<|z|^2, |y|<|z|, 0<z<1 \}$ and $f: D\rightarrow R, f(x,y,z)=2x+2y+z^3$. I would like to calculate the 3-dimensional volume of $G := \{(x, f(x)) | x \in ...
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2answers
45 views

What is the union of all the tangent plane at every point of a sphere?

Let $S = \{x \in \mathbb{R}^3: ||x||_2 = 1\}$ Then pick a point on $S$. The tangent space to the point is the plane that is perpendicular to the vector from origin to that point. What is the ...
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0answers
15 views

What is the topology of uniform convergence in this case of $P=P(x_{0},M)$ of all paths in $M$ starting at $x_{0}$?

The following definition I found it in a text on Lie groups: Let $M$ be a connected smooth manifold and $x_{0}\in M$. A path in $M$ starting at $x_{0}$ is a continuous curve $\gamma ...
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1answer
44 views

Lenght of the curve in Riemannian metric.

Let $M^{k}$ a submanifold, $h:U\to M^{k}$ a chart, and $\gamma:[a,b]\to h(U)\subset M^{k}$ a curve in $M^{k}$. Represent the curve in coordinates $(h,U)$ as ...
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0answers
26 views

Why is the dimension of a real submanifold uniqely determined?

Let $X$ denote a finite dimensional normed space. A non empty set $M \subset X$ is called $d$-dimensional differentiable submanifold of $X$, if for all $a \in M$ there exists an open neighborhood ...
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1answer
29 views

Show that $M$ is a 2-dimensional submanifold.

Let $f:\mathbb{C}\to\mathbb{C}$ be a complex polynomial $f=a_{0}+a_{1}z+...+a_{n}z^{n}$ without double zeroes. Consider for every natural number $k\geq 2$ the set $$M=\{(z,w)\in\mathbb{C}^{2}: ...
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1answer
18 views

Given a one-t0-one function f that maps M onto an arbitrary set A, prove there is a unique way to make A a manifold s.t. f becomes a diffeomorphism.

I'm really unsure of how to proceed, I've drawn a picture and can understand the general setting but don't know how to actually prove it.
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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
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3answers
92 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 ...
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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 ...
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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 ...
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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 ...
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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
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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
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2answers
106 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
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0answers
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 ...
0
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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
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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 $\int_A \alpha \in \mathbb{Z}$, ...
3
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0answers
61 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
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0answers
68 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
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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 ...
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1answer
38 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 ...
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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
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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
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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 ...
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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 ...
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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?
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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 ...
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1answer
51 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
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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 ...
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0answers
19 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 ...
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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
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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
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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 ...
6
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3answers
163 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
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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 ...
5
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1answer
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
votes
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
votes
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 ...
10
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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}} ...
7
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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.) ...