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

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1answer
53 views

Map from a manifold to $[0,1]$

I am looking through a practice exam to prepare for an upcoming final and I am having through with this question. Question: Let $M$ be a manifold, $p \in M$, and $U \subset M$ an open set containing ...
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1answer
77 views

Why $\Sigma$ is minimal, if $\frac{d}{dt} |_{t=0} \mathrm{Area}(\Sigma_t)=0$?

In this work http://arxiv.org/pdf/1204.2883v1.pdf Martin Li claimed that $\Sigma\subset M$ is minimal and $\Sigma$ meets $\partial M$ orthogonally along $\partial \Sigma$ if, only if, $$0 = ...
2
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1answer
278 views

Is there a compact contractible manifold?

Does there exist a compact connected manifold (without boundary), that has a trivial homotopy type?
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1answer
565 views

How to show $\omega^n$ is a volume form in a symplectic manifold $(M, \omega)$?

I have the following problem: Let $(M, \omega)$ be a symplectic manifold. How can I show $$\omega^n=\underbrace{\omega\wedge \ldots\wedge \omega}_{n-times},$$ satisfies $\omega^n(p)\neq 0$ for all ...
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2answers
63 views

Show that f is covering map and find covering tranformation group

Prove that $f:\mathbb{R}^2\to T^2$ defined by $f(x,y)=(e^{2\pi i x},e^{2\pi iy})$ is covering map and also find covering tranformation group$=\{g:\mathbb{R}^2\to\mathbb{R}^2\mid g$ is diffeomorphism ...
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2answers
87 views

Understanding the proof for: $d(f^*\omega)\overset{!}{=}f^*(d\omega)$

Consider this Proposition: Let $U\subset\mathbb{R}^n$ and $V\subset\mathbb{R}^n$ be open sets and $\phi:U\to V$ be differentiable. For all $k\in\mathbb{N}_0$ and $\omega\in \Lambda^k(V)$ it is true ...
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0answers
550 views

When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
3
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1answer
146 views

Tietze–Urysohn's lemma in $\mathbb{R}^n$

Let $F_1$ and $F_0$ be closed subsets in $\mathbb{R}^n$, $F_0\cap F_1=\varnothing$. How to build a $C^{\infty}$- function $f:\mathbb{R}^n\to \mathbb{R}$, such that $f|_{F_1}=1$, $f|_{F_0}=0$ and ...
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0answers
40 views

Formalizing a proof using Germs to define a linear and injective map of the Algebraic Tangent of a manifold

I am trying to show that for X being an n-dimensional manifold and Y a k-dimensional manifold, U an open set and both $Y,U \subset X, p\in U$ there is a natural and well defined and injective map ...
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1answer
43 views

help me in trace of following proposition

In a paper an author proved the following proposition Please help me in trace proof of following proposition Proposition: let $f$ be a homeomorphism of a connected topological manifold $M$ with ...
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2answers
146 views

Mayer-Vietoris for $\mathbb C\mathbb P^n$?

Does anyone have any idea of how calculating the De Rham cohomology $H^k(\mathbb C\mathbb P^n)$ of the complex projective space using Mayer-Vietoris?
2
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0answers
218 views

Manifolds: A definition of the Gradient an Algebraic Tangent vector over charts, how to show equivalence?

Let X be an n-dimensional differentiable manifold and $p \in X$ . Let $(U, h, V )$ for X around p with coordinates $(x_1 , . . . , x_n )$ in V , and let $v_i , i = 1, . . . , n$ , be the basis of ...
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2answers
113 views

basic doubt about topological manifold

In his book "Introduction to Smooth Manifolds", J.M. Lee defines a topological manifold to be a second countable, Hausdorff space with every point having a neighbourhood homeomorphic to an open subset ...
5
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1answer
97 views

Smooth homotopic maps and closed forms..

does anyone have any idea for showing the following: Let $f_0, f_1:M\rightarrow N$ smooth homotopic maps between the manifolds $M$ and $N$. Suppose $M$ is compact with no boundary. Show that for every ...
6
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1answer
194 views

Is a continuous map between smoothable manifolds always smoothable?

Let $X$ and $Y$ be topological manifolds and $f:X\to Y$ a continuous map. Suppose $X$ and $Y$ admit a differentiable structure (at least one). My question: is it always possible to choose a ...
6
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2answers
277 views

Ideal of smooth function on a manifold vanishing at a point

I'm trying to prove the following lemma: let $M$ be a smooth manifold and consider the algebra $C^{\infty}(M)$ of smooth functions $f\colon M \to \mathbb{R}$. Given $x_0 \in M$, consider the ideals ...
4
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1answer
122 views

Prove that the circle $S^1$ is not the boundary of any compact manifold with boundary in $\mathbb R^2-{(0,0)}$

Suppose it were, then define a 1-form $w:=\frac{1}{x^2+y^2}(-y\,\mathrm dx+x\,\mathrm dy)$. Firstly , I try to evaluate $\int_{S^1}w$ by two ways . Firstly, let $F\colon[0,2 \pi]\to S^1$ defined by ...
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1answer
61 views

if $X$ is a vector field how can I find $Y$ such that $[X,Y]=0$?

Suppose I am given a holomorphic vector field $X$ over a complex manifold $M$. To simplify this we can suppose that $X$ is a holomorphic vector field in $\mathbb{C}^n$ for $n=2$ or $n=3$. How can I ...
3
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1answer
73 views

Axiomatizing oriented cobordism

According to the nLab entry for abstract cobordism categories, the natural way of axiomatizing the relation of two oriented manifolds being cobordant is the following: Definition 1 Two objects ...
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1answer
421 views

proof of stokes theorem

I don't understand the "idea" of the following proof, as well as some of the steps. As i'm not sure about its "ways", im not editing it much and as such it might be in the wrong order. My sincere ...
2
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1answer
100 views

Diagonal Inclusion Map of a manifold $X$

This question actually comes from the question I asked before: Derivative map of the diagonal inclusion map on manifolds And I repeat it as follows: Let $f: X\longrightarrow X\times X$ be the ...
17
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2answers
671 views

The “Easiest” non-smoothable manifold

In 1960, Kervaire found the first example of a PL-manifold which does not admit a smooth structure. Since then, I understand that there are many examples of non-smoothable manifolds that can be built. ...
4
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1answer
202 views

Horn and spindle tori

I was trying to prove that the horn torus and the spindle torus are not manifolds by definition(locally diffeomorphic to some Euclidean space.). I have no idea how to do this, but I attempted it in ...
6
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1answer
268 views

Derivative map of the diagonal inclusion map on manifolds

I was trying to work through a problem(#10 of $\S$1.2) in Guillemin and Pollack's book $\textit{differential topology. }$ The problem is given as follows. Let $f: X\longrightarrow X\times X$ be ...
2
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1answer
186 views

Finding a direct basis for tangent space of piece with boundary of an oriented manifold.

I have the following definition (from Hubbard's vector calculus book) for an oriented boundary of piece with boundary of an oriented manifold: Let $M$ be a $k$ dimensional manifold oriented by ...
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1answer
106 views

$[M,\mathbb CP^\infty]=[M,\mathbb CP^2]$ where $M$ is a smooth closed orientable 3-manifold!

Prove the above result, where $[X,Y]$ means the set of all homotopy classes of maps from X to Y, two topological spaces. I have answered it below.
6
votes
1answer
129 views

Tangential Space of a differentiable manifold is always $\mathbb R^n$?

Let $\mathcal M$ be a differential manifold with a point $p$. Let U be an open set, $p\in U$, on $\mathcal M$ and let $\phi,\psi:U\to \mathbb R^n$ be a charts on $\mathcal M$. I'm having diffculties ...
3
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0answers
33 views

How to tell computationally that a volume of points constitutes a manifold

Suppose that a space ${\mathbb R}^{r}$ contains a set of points which we want to consider as enclosing a volume within the space, or perhaps a volume in a submanifold (e.g., the sphere $S^{2}$ within ...
3
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1answer
232 views

Algebraic Tangent Space and Vector - an intuitive understanding?

In this question I am looking for help in understanding the Algebraic Tangent vector and what the difference is between it and the "regular" Tangent vector. A "differentiable function" near p is a ...
0
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1answer
176 views

Geometric Tangent Vectors - looking for and understanding of and what the point is.

The problem that I am having is that I am having quite a hard time understanding the ideas of Geometric Tangent vectors and why they are even needed - I mean one already has the usual "Tangent" of a ...
5
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2answers
262 views

Injectivity of a map between manifolds

I'm learning the concepts of immersions at the moment. However I'm a bit confused when they define an immersion as a function $f: X\rightarrow Y$ where $X$ and $Y$ are manifolds with dim$X <$ ...
2
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0answers
76 views

Fundamental Group, Piecewise Smooth Curves, Consevative Fields

Let M be compact Riemannian manifold. X be a vector field on M. I believe that work done by X any piecewise closed curve is zero, iff, the same is zero for a particular finite set of loops. I believe ...
3
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0answers
57 views

Maps between total spaces of holomorphic vector bundles

I am wondering what is possible and what is not possible regarding maps between the total spaces of holomorphic vector bundles. Let me outline a situation that is a bit more concrete, to help focus ...
9
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3answers
655 views

How do I know when a form represents an integral cohomology class?

Suppose $M$ is an $n$-dimensional manifold, and $\omega \in \Omega^p(M)$ is a closed $p$-form. Moreover, assume that $d\omega = 0$, so that it represents a de Rham cohomology class. I would like to ...
2
votes
2answers
110 views

Vector Field Generating Variation Along Curve

I'm learning a proof of the fact that length extremising curves are geodesics of the Levi-Civita connection, and have found something I don't understand. The argument states the following. Suppose ...
13
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2answers
230 views

Topological manifolds (dimension)

I am taking an introductory course to topology and the professor defined a topological manifold of dimension $n$ if it is hausdorff and if for every point $x$ there exists an open set $U$ around $x$ ...
3
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1answer
64 views

Question about diffeomorphism

Here is an assignment problem: $f:\mathbb{S}^2 \longrightarrow \mathbb{S}^2$ is smooth and surjective. Prove $\exists$ open subset $ U $ of $\mathbb{S}^2$, such that $f|_U$ is a diffeomorphism. I've ...
2
votes
1answer
93 views

Lebesgue covering dimension of a manifold

I have found many sources saying that the Lebesgue covering dimension of a (topological or smooth) manifold is the same as the dimension of the manifold. Does anyone know where I can find the proof?
4
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0answers
132 views

Differentiable manifolds, Serge Lang

I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His ...
10
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1answer
229 views

Invariant submanifolds

Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
2
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2answers
93 views

Smooth maps on a manifold lie group

$$ \operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\ \begin{align} &n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\ &n = 2, \operatorname{GL}_n(\mathbb ...
3
votes
1answer
116 views

question from hatcher basic 3 manifolds

The question is: why should a homologically trivial embedded sphere in a simply connected (not necessarily compact) 3 manifold M bound a compact 3 manifold embedded in M? I had this problem reading ...
3
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2answers
173 views

Attaching two manifolds along their boundary

I have a question about a proof in John Lee's Introduction to Topological Manifolds. Suppose $M$ and $N$ are two topological $n$-manifolds with nonempty boundary (for reference, the definition I am ...
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1answer
267 views

How to directly show that Figure 8 injective immersion is not a monomorphism

I'm working in the category of smooth manifolds. The injective immersion that takes the open unit interval $(0,1)$ to the figure 8 is a well-known example of an injective immersion that is not an ...
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2answers
149 views

The euclidean space $\Bbb R^n$ is orientable as a manifold.

I know that The euclidean space $\Bbb R^n$ is orientable as a manifold. I think that it is orientable because it has a nowhere vanishing $n$-form. But I am not sure. Please can you explain ...
3
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1answer
1k views

Differential Geometry Video Lectures

I know there's a similar question here, however since what I found there wasn't what I was looking for I thought on creating a new question. I'm studying Differential Geometry through Spivak's book "A ...
2
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0answers
62 views

Stoke's theorem application to curl theorem. I did. Please can you check it?

Now, I need to apply stoke's theorem to curl theorem. My teacher gave a hint. Accourding to the hint, I accept $w=Pdy∧dz +Q dz∧dx + R dx∧dy$ $\in Ω^2(M)$ $dim(M)=2$ M is the subset of $\Bbb ...
3
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0answers
631 views

Real projective space is Hausdorff

I could not understand the proof of thıs proposition can you help me and give clear explanation.Just can you say how we have (n+1)x2 matrix?? This prove is correct or I need to add something ?? ...
2
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1answer
101 views

Real Projective Space

Corollary $\bf7.15.$ The real projective space $\mathbb{R}P^n$ is second countable. How I can prove this corollary 7.15.I consider that ı can apply the corollary 7.10.But I could not can you help ...
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1answer
46 views

Locally finite or not

I am tryıng to learn locally finite and can you give an explanation for my green writing please thank you