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

learn more… | top users | synonyms (1)

4
votes
2answers
55 views

signature of the topological manifold $M= \mathbb{C}P^6\times \mathbb{C}P^6$ (using homology, cohomology)

I want to prove that the signature $\operatorname{sig}(M)$ of the topological manifold $M= \mathbb{C}P^6\times \mathbb{C}P^6$ is nonzero. First of all, $M$ is a compact 24-dimensional manifold ...
1
vote
1answer
42 views

Two figures, one is not a manifold, but the other is a manifold, but they look both as suffering from the same deficiencies for not being a manifold

I am looking video lectures of F. Schuller about space-time geometry, in particular about manifolds. In it, right at the beginning after introducing manifolds he gives a non-example, i.e. he says that ...
1
vote
0answers
24 views

How does an almost complex structure on a manifold induce an orientation?

I have read that given a smooth even dimensional manifold $M$ with an almost complex structure $J$, then $M$ is orientable and there is a canonical choice of orientation. Why is this the case? How ...
3
votes
2answers
63 views

Lee, Introduction to Smooth Manifolds, Change of Coordinates

In all versions of John M. Lee's Introduction to Smooth Manifolds, he claims that $$\left(\psi\circ\varphi^{-1}\right)_*\left.\frac{\partial}{\partial ...
2
votes
1answer
47 views

How to define CW-complex structure on cubic surface in $CP^3$?

I have read roughly this blog and I have following question. I changed my original question to following. How to define CW-complex structure on cubic hypersurface $M$ in $\mathbb CP^3$ defined by ...
2
votes
0answers
41 views

Calculation of extrinsic curvature

I asked this question first on physics.SE but I got no complete answer so I thought maybe someone here could help. I'm trying to understand how to derive the extrinsic curvature (in order to ...
0
votes
0answers
25 views

Maximal flow of a linear vector field in $M_n(\mathbb R)$

How to determine the maximal flow $Φ^X$ of the linear vector field $X_A(x) := Ax$. Where $A\in M_n(\mathbb R)$?
3
votes
1answer
40 views

If $M$ is a manifold, then $\partial(\partial M)) = \emptyset.$

If $M$ is a manifold, then $\partial(\partial M)) = \emptyset.$ I've searched this question here and I did not find any solution. I know that this problem is equivalent to show that $\partial(\partial ...
1
vote
1answer
31 views

Whether there exists some compact manifold whose continuous map must have a fixed point?

Whether there is some compact manifold $M$ such that $f:M\rightarrow M$ is continuous $\Rightarrow f$ has a fix point. I mean that any continuous map from $M$ to $M$ must have a fixed point. I ...
1
vote
1answer
39 views

Two vector bundles over same base manifold $X$

What are two vector bundles over the same base manifold $X$ which are isomorphic as vector bundles in the general sense, but not isomorphic over $X$? (That is to say, this would demonstrate that there ...
2
votes
1answer
59 views

Is it possible to have a connected manifold that is a double cover of a 2-sphere?

I have come up with a branched covering, but it necessarily has two branch points. From that I'm assuming that it can't be done, possibly related to the hairy ball theorem, but I don't know how to ...
0
votes
0answers
27 views

unique differentiable atlas making $F$ diffeomorphims

Let $M$ be a differentiable manifold and $F:M\to N$ be a bijective mapping. If $M$ is a topological space, then $N$ admits a unique topology making $F$ a homeomorphism; it is easy prove that this ...
0
votes
0answers
31 views

Show that $\mathbb{PR}^1$ is diffeomorphic to $S^1$

I know how to construct the charts. I will have: $$\phi_i(x_1,x_2) : U_i \to \mathbb{R}$$ where $Y_i \subset \mathbb{RP}^1$ defined by: $U_i :=\{(x_1,x_2) : x_i > 0\},$ $\phi_1(x_1,x_2) = ...
0
votes
1answer
30 views

dimension of smooth connected manifold

Let $M^n$ be a smooth, connected $n$-manifold. By definition, this means that every point $p\in M$ has a neighborhood $U$ that is homeomorphic to some open subset of $\mathbb{R}^n$. Now let us assume ...
0
votes
1answer
20 views

Function times a vector field (one-forms)

I am reading an introduction to differential geometry in the book "Classical Dynamics: a contemporary approach" by Eugene and Saletan (page 135). Consider a manifold $Q$ of dimension $n$. They define ...
2
votes
1answer
55 views

Change of coordinates between charts

Let $M$ be a differentiable manifold, $(U, \phi)$ and $(V,\psi)$ two coordinate charts and $p$ a point of $M$. Let $\{ \frac{\partial}{\partial \phi_{1}} (p), \ldots, \frac{\partial}{\partial ...
2
votes
2answers
49 views

When does a codimension 1 submanifold admit a transverse vector field?

I'm having some trouble with the following problem, which comes from a released qualifying exam: Assume that $N \subset M$ is a codimension 1 properly embedded submanifold. Show that $N$ can be ...
3
votes
2answers
88 views

Exist vector field having only finitely many zeros, all lying in open set of compact connected manifold?

Let $U$ be any open set on the compact connected manifold $X$. Does there exist a vector field having only finitely many zeros, all of which lie in $U$?
0
votes
0answers
44 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
votes
0answers
32 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 ...
1
vote
1answer
28 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 ...
1
vote
1answer
43 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 ...
1
vote
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 ...
2
votes
1answer
43 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 $(X_0,Y_0)$?
1
vote
0answers
54 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?
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
25 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 ...
0
votes
2answers
41 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 ...
1
vote
0answers
14 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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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}: ...
0
votes
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.
1
vote
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
90 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
49 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
votes
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
59 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
47 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
vote
0answers
40 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" ...