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)

1
vote
1answer
26 views

What are the charts that make up an atlas for the long line?

This question is prompted while I was working through the MIT OCW (Massachusetts Institute of Technology, Open CourseWare) for 18.965, ``Geometry of Manifolds,'' in its Lecture 2, ...
0
votes
0answers
34 views

Topological boundry on orientable manifold

Let $X\subset \mathbb R^n$ be a non-empty $(n-1)$-dimensional sub-manifold for some $n\geq 2$. Assume there exists some open $U\subset\mathbb R^n$ with $x\subset U$ and a differentiable function ...
0
votes
1answer
51 views

Start of the proof of the Poincaré Lemma

Let $\hat{\Phi}:U \rightarrow U$ be a one-parameter family of diffeomorphisms defined for $\ 0 < t \leq 1$. Let $\beta \in \Omega^k(U) $ be a closed k-form. Suppose that $\hat{\Phi}^{*}_1 \beta = ...
0
votes
0answers
28 views

Converting a word problem to algebra

This is a forming of an equation, which I haven't been able to get my head around. I have a worked solution to this problem. Question: For $x\in\mathbb{R}^m$ and $\epsilon>0$, show that ...
0
votes
1answer
37 views

Local coordinates on a product of two manifolds.

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. I think that a local coordinate on $X \times Y$ is $(U \times V, x_1 ...
1
vote
0answers
38 views

Line with two origins is a manifold but not Hausdorff

The line with two origins is $(\mathbb{R} \times \{0,1\})/\sim$ where $(x,0)\sim(x,1)$ for $x\neq 0$. I can see that it is not Hausdorff, since we cannot separate the points $(0,0)$ and $(0,1)$. ...
10
votes
2answers
218 views

Are compact complete geodesics closed?

Let $(M,g)$ be a compact Riemannian manifold. Is there an example of a geodesic $c:\mathbb{R}\to M$ s.t. $c(\mathbb{R})$ is compact, $c$ is NOT periodic (i.e. be NOT a closed geodesic) ?
1
vote
1answer
31 views

A Smooth map homotopic to a constant map

Q: Let $M^{k}$ be a smooth compact $k$-manifold and let $F:M \rightarrow S^{n}$ be a smooth map, where $n>k$. Prove that $F$ is homotopic to a constant map. Proof: Since $n>k$, by Sard's ...
1
vote
1answer
36 views

Topological structure of the Manifold valued functions

$M$ is a Riemannian manifold. What condition on $M$ for $\mathcal{C}_{[a,b]}(M)$ (the set of continuous functions of the real interval $I=[a,b]$ to $M$) to be a polish space ? For which topology ? Is ...
1
vote
0answers
19 views

Proof of Triangulation Theorem for 1-Manifolds

While I am reading "Introduction to Topological Manifolds" by John M. Lee, I come to see the following paragraph in the proof of Theorem 5.10 pp. 102. Note that Int$\ e\cap\ $Int$\ e'$ is open in ...
4
votes
1answer
38 views
0
votes
0answers
22 views

Classification of surface with 18-gon planar diagram

For starters, this is a problem from L. Christine Kinsey's "Topology of Surfaces." The problem is to classify the surface using cut and paste arguments on polygons. However, between my limited ...
2
votes
3answers
82 views

Constructing a vector bundle using Vector bundle construction lemma

Given are: an open cover of $\{U_\alpha\}_{\alpha\in A}$ of a smooth manifold $M$. smooth maps $\tau_{\alpha\beta}\colon U_{\alpha}\cap U_{\beta}\rightarrow \text{GL}(k,\mathbb{R})$ with ...
0
votes
1answer
54 views

Sobolev space on closed surfaces

I was wondering if anybody here knows how the Sobolev space $H^2(\mathbb{S}^2)$ is defined? I.e. I want to integrate on this space with respect to the surface measure, but since this not the canonical ...
0
votes
0answers
49 views

How to define an action of vector field on $C^{\infty}(M)$?

Let $M$ be a manifold. Let $\hat{X}$ be a vector field on $M$. How to define an action of $\hat{X}$ on $C^{\infty}(M)$? Thank you very much.
1
vote
1answer
27 views

How to define the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$?

I read a paper and on page 9, the paragraph before Proposition 5, it is said that let the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$. How to ...
6
votes
1answer
75 views

If it looks like a solid torus, walks like a solid torus, and quacks like a solid torus, is it a solid torus?

If an orientable 3-manifold $M$ has boundary the torus $S^1\times S^1$ and deform retracts to a solid torus $S^1\times D^2$, is it necessarily homeomorphic to a solid torus? Equivalently, if the ...
0
votes
1answer
25 views

Is this subset of a smooth manifold, a submanifold?

I'm not much informed about manifold but I should answer some questions about it. Based on the definition I have written an answer for the following question but I feel there is something wrong with ...
2
votes
1answer
36 views

Whitney's Embedding

The Whitney embedding theorem says that any smooth manifold of dimension $n$ may be embedded in $R^{2n}$. I am just beginning to study differential geometry for application to physics (general ...
7
votes
3answers
164 views
0
votes
1answer
37 views

Hypersurface is not curved, normal vector

Let $S$ be a hypersurface in $\mathbb{R}^n$. Is there a simple way to say that $S$ is flat by describing the normal vectors on $S$? Like $S$ is flat if the normal vectors on $S$ are all identical..
1
vote
2answers
57 views

Smooth homotopy

Let $M,N$ be manifolds. Suppose that $f_0, f_1:M\stackrel{C^\infty}\to N$ are homotopic, i.e. there exists a continuous mapping $f:M\times[0,1]\to N$ s.t. $f(x,0)=f_0(x)$, $f(x,1)=f_1(x)$. Then is ...
0
votes
1answer
35 views

0th-order differential operator vs. 1st-order differential operator on a vector bundle $(E, \pi,M)$

Consider a vector bundle $(E,\pi,M)$. A 0th-order differential operator on $E$ is a $C^\infty(M)$-linear endomorphism $\Gamma E\rightarrow\Gamma E$. $\Gamma E$ is the set of sections on $M$. A ...
1
vote
0answers
29 views

Calculating Integral Submanifolds

I have the vector fields $v_{1} = x \partial_y - y \partial_x + z \partial_w - w \partial_z$ and $v_{2} = z \partial_x - x \partial_z + w \partial_y - y \partial_w$ on $S^{3} \subset \mathbb{R}^4$. I ...
2
votes
2answers
181 views

Integration of a 2-form

$\textit{What is}$ $\int_C{\omega}$ $\textit{where}$ $\omega=\frac{dx \wedge dy}{x^2+y^2}$ $\textit{and}$ $C(t_1,t_2)=(t_1+1)(\cos(2\pi t_2),\sin(2\pi t_2)) : I_2 \rightarrow \mathbb{R}^2 - ...
2
votes
0answers
28 views

Intersection of invariant subsets of a local group action

I don't understand some facts about invariant subsets of a local group action. Basically (to save you reading definitions) local actions are germs of partial actions which in turn are just like ...
0
votes
0answers
11 views

how can I do an implicit integration by “picking” the right function?

Supposing I have a function $f(x,y) = y/x$. The relationship of $y$ and $x$ is determined through an implicit unknown function $g(y,x)=0$, so that there is a unique path from two points. How can I ...
0
votes
1answer
87 views

Solving for Center Manifold with Parameter

I have a system of ODEs given by $$\frac{dX}{d\tau}=\beta X\left(1 - \frac{X+Y}{N}\right)$$ $$\frac{dY}{d\tau}= Y\left(1 - \frac{X+Y}{N}\right)$$ where $\beta $ is a parameter. How should I ...
0
votes
1answer
38 views

Immersion locally injective?

I was wondering whether any immersion is locally injective?-This definitely sounds natural and I guess that it is true, as the derivative of an immersion is globally injective. Thus, there cannot be ...
2
votes
1answer
39 views

What are the elements in $\Gamma(\Lambda^2 TM)$?

In the lecture notes, Proposition 1.19 on page 9, it is said that on every Poisson manifold there is a unique bivector field $\Pi \in \Gamma(\Lambda^2 TM)$ such that $$ \{f, g\} = \langle \Pi, df ...
2
votes
0answers
48 views

Embedding vs embedding as a closed subset

A version of Whitney's Theorem state that any $n-$dimensional manifold can be embedded in $\mathbb{R}^{2n+1}$ as a closed subset. Is there an example of $n-$manifold which can be embedded in ...
3
votes
1answer
186 views

Rank Theorem proof

Let $\phi: M \to N$ be an immersion from smooth manifold $M^m$ into $N^n$ ($\dim M = m$ and $\dim N = n$). Prove there exists smooth charts $(U,h)$ in $M$ with $p \in U$, $h(p) = 0$, and $(V,g)$ in ...
5
votes
1answer
89 views

One-forms in differentiable manifolds and differentials in calculus

Suppose that we have this metric and want to find null paths: $$ds^2=-dt^2+dx^2$$ We can easily treat $dt$ and $dx$ "like" differentials in calculus and obtain for $ds=0$ $$dx=\pm dt \to x=\pm t$$ ...
1
vote
0answers
34 views

Fundamental Group of an n-dimensional manifold with finite k punctures

Problem: Show that $\pi_1(M\setminus${k points}$) = \pi_1(M)$, where $M$ is an n-dimensional manifold ($n\ge 3)$ and k is a positive integer. In class, we went over the proof for the above ...
3
votes
1answer
60 views

Infinite Genus Riemann Surfaces

I want to show that every infinite genus Riemann surface $M$ has a proper closed subset such that, $M^*\setminus E$ ($M^*$ is the one-point compactification of $M$) is connected and locally ...
0
votes
0answers
46 views

Distance function under diffeomorphism of manifolds

community, I am concerned with measuring distances of systems under diffeomorphisms. Concrety, I consider a smooth diffeomorphism $\varphi: M\rightarrow N$ from the smooth differentiable manifold ...
1
vote
1answer
50 views

Differentiable Manifold minus point not compact

Suppose $X$ is an $n$-dimensional for differentiable manifold for $n \geq 1$: in our definition this is a second countable Hausdorff space with a maximal differentiable atlas. If $p \in X$ is a ...
4
votes
1answer
103 views

How can I visualize what open sets “look” like in unfamiliar topological spaces?

The question is extremely general, but I do have a specific case I'd like to look at, and I'm hoping that some combination of specific pointers and general advice will help me out. Consider the ...
1
vote
1answer
49 views

Diffeomorphism between the Grassmannian manifolds $\mathbf{Gr}(n,k)$ and $\mathbf{Gr}(n,n-k)$.

This seems to be a common exercise question, however I am having trouble with it. The hint is to use a map that associates the k-plane to its orthogonal complement. But I have not been able to show ...
0
votes
1answer
32 views

Component formula for pullback of one forms

Should it not be $\Big(F'(x)v \Big)^j = \frac{\partial F^j}{\partial x^k }(x)v^k$? Then also how is $F^*dy^j = \frac{\partial F^j}{\partial x^i}dx^i$ derived? I cannot what $\beta_j$ has been ...
3
votes
1answer
117 views

How to calculate differential forms on $S^2$ parameterized by stereographic projection?

Suppose that we have the stereographic mapping $\varphi: \mathbb{R}^2\to M$ where $M=S^2-\{(0,0,1)\}$. I've already found that the stereographic parametrization of $S^2-\{(0,0,1)\}$ is given by: ...
2
votes
1answer
58 views

On the definitions of $n$-manifold etc.

I'm doing 3rd year undergraduate geometry (an introductory subject), and we've been given formal definitions of terms like "$n$-manifold" and "smooth $n$-manifold." However, I tend to think about ...
3
votes
0answers
21 views

pontriyagin class of quaternionic vector bundle

Let $\xi^{\mathbb{H}}$ be a quaternionic vector bundle over $X$. How to define the Pontriyagin class of $\xi^{\mathbb{H}}$ efficiently? Of course we can let $(\xi^{\mathbb{H}})_{\mathbb{R}}$ be the ...
2
votes
1answer
46 views

function of class C ^ 1 on manifolds

Let $M$ be a differentiable manifold with finite dimension $ m $. Let $ f:M\rightarrow M $ a function of class $C^1$. I have a doubt about what this implies (1) or (2): $x \in M \rightarrow D_xf ...
5
votes
2answers
106 views

frustrating experience about differential geometry

I am felling rather frustrated now, after taking a long time to study differential geometry, but with little progress... Indeed my major is mainly numerical analysis. I am studying modern geometry, ...
1
vote
1answer
51 views

Smooth map on submanifold

Is the following true? Let $M$ be a differential manifold and $f : M \to M$ be a smooth map. If $N$ is a submanifold of $M$ and $f(N)\subset N$ then the restriction $f|_N : N \to N$ is smooth.
1
vote
1answer
44 views

Is rectangle manifold with boundary

Is a closed rectangle a 2d manifold with boundary? It seems like the corners don't have neighborhoods homeomorphic to the Euclidean half space?
1
vote
1answer
57 views

Meaning of the adjoint representation of a Lie group

The adjoint representaion of $G$ is a homomorphism $ad_{a}:g \rightarrow aga^{-1}$, $a,g \in G$, what is the meaning of this? Now if we identify $T_{e}G$ with $\mathfrak{g}$ we have the adjoint map ...
3
votes
1answer
60 views

Integrating tensors on manifolds

When/how can you integrate tensors on manifolds and what does it mean? I imagine that line integrals of tensors make sense when you have a connection, since you can uniquely parallel transport all ...
1
vote
2answers
38 views

How to show $X=\{p\in M: \textrm{ker}(df(p))=\{0\}\}$ is open in $M$?

Let $M$ and $N$ be two smooth manifolds and $f:M\longrightarrow N$ a $C^\infty$ map. We say $f$ is an immersion at $p\in M$ if $df(p):T_pM\longrightarrow T_{f(p)}N$ is injective. How can I show the ...