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

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1answer
75 views

Reference Request: topological h-cobordism theorem in higher dimensions

The h-cobordism theorem is true in the topological and in the smooth category in dimensions $\ge 6$. (By "dimension, I mean the dimension of the ambient cobordism instead of the dimension of the ...
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2answers
95 views

Two definitions of integral on boundary $\int_{\partial\Omega}f$?

I have seen two definitions of an integral of a function $f:\partial\Omega \to \mathbb{R}$ from the boundary of an open set $\Omega \subset \mathbb{R}^n$ where the domain is Lipschitz. 1) ...
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1answer
58 views

Bounding an integral on boundary of Lipschitz domain

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain with bounded boundary $\Gamma.$ So $\Gamma$ is a hypersurface of dimension $(n-1)$. I want to show that $$\int_\Gamma ...
0
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1answer
42 views

Lipschitz domain and surface measure

Let $S$ be the boundary of a Lipschitz domain $\Omega$. We know it has a surface measure $\mu$. Can we write $d\mu = f(x)dx$ with $f$ explicity given in terms of the Lipschitz maps that make up the ...
3
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0answers
72 views

Homological Interpretation of the intersection number

Let $M^r$ and $N^s$ be smooth submanifolds of the smooth manifold $V^{r+s}$ and $M,N,V$ compact, connected and orientable. The Thom-Isomorphism together with the Tubular Neighbourhood Theorem gives me ...
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0answers
44 views

Seifert manifolds

Seifert fiber space is a PFB. The theorem states that every principal fiber bundle (PFB) admits a connection form, so how can we define the connection 1-form on it? Or how can I find a book or article ...
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1answer
91 views

$df$ vanish in a compact manifold in at least 2 points

I need to prove that if $M$ is a compact manifold and $f$ is a smooth function in $M$, then $df$ vanish in at least 2 different points of $M$. I don't know where to start. Any suggestion will be ...
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0answers
21 views

homology of face links of a triangulated manifold

In a triangulation of a general topological space, we can define a face link (for any face in the triangulation). Intuitively, this is a kind of "$\epsilon-$sphere" in metric space. In chapter 3.8 ...
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2answers
65 views

What is the equivalent of Delaunay tringulations in high dimensions?

For 2D manifolds, Delaunay triangulation is a very useful tool for coarse graining. It has the nice property that in the flat/euclidian manifold case, it reduces to a 2D simplicial tesselation of the ...
5
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1answer
104 views

Is there any way to define differentiablity without any reference to the Euclidean space?

We define metric spaces based on the properties of the real numbers $\Bbb{R}$. In the same spirit we define smooth manifolds. But there is a more general and elegant way to formulate our intuition of ...
2
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0answers
90 views

Definition of linking number for disjoint submanifolds of the sphere- A problem from Milnor's book

In problem number 13 of Milnor's 'Topology from the Differentiable Viewpoint', the linking number for two compact boundary-less manifolds $M,N \subset \mathbb{R}^{k+1}$ of dimensions $m,n$ such that ...
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0answers
175 views

Top de Rham cohomology

I just realized that I never really understood why $H_{dR}^n(M, \mathbb{R}) = \mathbb{R}$ if $M$ is compact and $H_{dR}^n(M, \mathbb{R}) = \{0\}$ if $M$ is not compact (provided that's true?). I'm ...
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0answers
47 views

On the topology of a Riemann manifold

Given a Riemann manifold $(M,g)$,the Riemann metric induces a topology on $M$ which given by $d(p,q)$=the shortest length between $p$ and $q$,it's a metric topology,and my question is:is this topology ...
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0answers
11 views

Trivial Lie Algebroid

Let $A\longrightarrow M$ be a vector bundle, $\mathfrak{g}$ a lie algebra and $A:=TM\oplus (M\times \mathfrak{g})$. I read I can define a Lie bracket on the space of section $\Gamma(A)$ as follows, ...
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0answers
41 views

Restriction of a Lie bracket on the space of section of a vector bundle..

Let $A\longrightarrow M$ be a vector bundle and $U\subseteq M$ an open set. Suppose I have a lie bracket on $\Gamma(A)$ such that if $\rho:A\longrightarrow TM$ is a bundle map then $$[a, fb]=f[a, ...
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1answer
24 views

What would be the space of section of the bundle $\mathfrak{g}\longrightarrow \{e\}$?

Let $\mathfrak{g}$ be a Lie algebra and $\pi:\mathfrak{g}\longrightarrow \{e\}$a vector bundle over a point. What would be the sections of this bundle?
2
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1answer
101 views

Why is partition of unity required in definition of Sobolev space on manfolds?

Why do we need to use $\phi_i u$ in the expression for the norm? Why not just $u$? The range of integration is over $R(x_i)$ anyway, so I don't understand why it is necessary. If you check Kendall, ...
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2answers
76 views

Why is this curve a topological manifold?

Why is $$M=\{(z_1,z_2)\in \mathbb{C}^2 \, |\,\, z_1^3-z_2^4=0 \}$$ a topological manifold? I understand for example why why $|z|=1$ is a topological manifold, since I can write every point as ...
0
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2answers
81 views

Orientability of surfaces in $\mathbb{R}^3$

I have a question: I'm currently reading a few things about orientability and understood this concept as the answer to the question: Given a surface and a unit normal vector field on it: Is there a ...
1
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1answer
52 views

Using transition maps as a comparison tool between charts on a manifold.

In the wikipedia article http://en.wikipedia.org/wiki/Chart_%28topology%29#Transition_maps we read A transition map provides a way of comparing two charts of an atlas. To make this comparison, we ...
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2answers
120 views

Is the cube with boundary and corners a manifold with boundary?

The definition of a n-manifold with boundary as I understand it, is that the manifold without boundary is an n-manifold, and the boundary is an (n-1)-manifold. Thus because the boundary of cube has ...
4
votes
1answer
104 views

The unsolved extension problem on manifolds.

I have been struggeling for quite a while with this problem: Let $M \subset \mathbb{R}^n$ be a compact $C^k-$ submanifold and $\phi_i: B_i(0) \rightarrow M$ be the associated set of charts ...
1
vote
1answer
141 views

How do 1-d compact submanifolds look like?

I was wondering whether it is true that every 1-d compact submanifold $\subset \mathbb{R}^n$ that is $C^1$ is a closed curve that is also $C^1$, cause I cannot think of more examples. Therefore, I ...
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0answers
23 views

Orientability of Ringed Space

Differential manifold can be defined in two ways. One definition is a topological space equipped with an atlas and transition maps. Another definition is a topological space equipped with a sheaf of ...
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1answer
46 views

a question about compact tangent bundle

I have a question about tangent bundles. Is there a compact tangent bundle? Or what conditions do we need to be sure that tangent bundle of a manifold be compact?
2
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2answers
48 views

Tangent bundle of sphere with $g$ handles

How can one show that tangent bundle $TM$ is not trivial if $M$ is a sphere with $g$ handles and $g \ne 1$?
3
votes
2answers
110 views

When can a manifold be curvature free?

Recall that in a Riemannian manifold (or pseudo Riemannian) there is always the unique Levi-Civita connexion that annuls the torsion. There are also manifolds (not needfully Riemannian) which are ...
2
votes
1answer
28 views

sufficient condition for being an integral factor

Let $ f: \mathbb {R}^m \rightarrow \mathbb {R}-\{0\} $ function $C^{\infty}$ class and $w$ a one-form $C^{\infty}$ class in $\mathbb {R}^m $. If $\alpha=w-\dfrac{1}{f}dx_{m+1} $ satisfies $\alpha ...
2
votes
1answer
63 views

Product of manifolds with boundary

If $M$ and $N$ are manifolds with boundaries and $\{(U_a,f_a)\}$ and $\{(V_a,g_a)\}$ are their respectives $C^r$ atlas, why $\{(U_a \times V_b,f_a \times g_b)\}$ isn't an $C^r$ atlas for $M \times ...
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2answers
105 views

An example of a smooth map between manifolds that is a topological embedding, but is NOT a smooth embedding.

I have been reading Lee's book on smooth manifolds and have come across the problem, An example of a smooth map between manifolds that is a topological embedding, but is NOT a smooth embedding. ...
1
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1answer
111 views

Questions about manifolds

This is a question from spivak and "proper" means the inverse image of any compact set of N is still compact. However, I can not find a suitable compact subset of N to use this property.
3
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2answers
226 views

Proof that the set is not a manifold

How to show that set: $M = \left\{(x,y,z): x^2+y^2+3z^3 = xy + 6z^{\frac{1}{3}}, z \neq 0\right\} \cup \left\{(0,0,0)\right\}$ is not a manifold? I know the problem is with point $(0,0,0)$. I think i ...
3
votes
2answers
80 views

Do there exist cancellable manifolds?

I do not know whether there exists a terminology for that property, but let us say that a closed manifold $C$ is cancellable if for every closed manifolds $M_1$ and $M_2$, $C \times M_1$ and $C \times ...
0
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1answer
38 views

If $\gamma : J \mapsto M$ is a smooth curve in a smooth manifold M, then $\gamma'(t) \neq 0$ $\forall t \in J$ iff $d\gamma$ is injective.

If $\gamma : J \mapsto M$ is a smooth curve in a smooth manifold M, then $\gamma'(t) \neq 0$ $\forall t \in J$ iff $d\gamma$ is injective. Here $J$ is just an open interval of $\mathbb{R}$ I'm just ...
1
vote
1answer
43 views

Existence of a nonzero vector to form

Let $ f: \mathbb {R}^m\times \mathbb {R}^m \rightarrow \mathbb {R}^m $ an alternate form of grade two. If $ m $ is odd, prove that there exists $ v\neq 0 $ such that $ f (u, v) = 0 $, for all $ u \in ...
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1answer
58 views

How prove that $\mathbb{CP}^2$ is compact? [closed]

How prove that $\mathbb{CP}^2$ is a compact manifold.
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0answers
39 views

An $n-1$ dimensional surface has $n$ dimensional measure $0$.

How does one show this? I was thinking that on an $(n-1)$ dimensional surface there a local homeomorphism to $\mathbb{R}^{n-1}$, which can be canonically embedded into $\mathbb{R}^n$, and it seems ...
5
votes
1answer
91 views

Volume of neighborhood of the curve

Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|<r\}$. How can i show ...
3
votes
1answer
48 views

1-form on $S^n$ with non-degenerate differential.

How it can be solved? Whether there is a $1$-differential form $w$ on $S^n$, such that $dw$ is non-degenerate on $T_aS^n$ for each $a \in S^n$?
4
votes
1answer
343 views

Euler characteristic is equal to self-intersection number of zero-section?

As I recall (from Guillemin and Pollack "Differential Topology") the Euler characteristic of a (for my purposes, compact and oriented) smooth manifold X is defined as $\chi(X)=I(\Delta,\Delta)$, ...
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1answer
40 views

How to show that the metric in the tangent space is independent from the chart you take?

I want to prove that for vectors $v_1,v_2 \in T_aM$ the euclidean length and distance is independent from the chart we are using, where $M$ is a submanifold in some $\mathbb{R}^n$ My problem is that ...
2
votes
1answer
99 views

Derivative of antipodal map between $n$-spheres

Let $S^{n-1}\subseteq \mathbb{R}^n$ denote the $(n-1)$-sphere $x_1^2+\ldots+x_n^2=1$. Let $f:S^{n-1}\rightarrow S^{n-1}$ be the map $f(x_1,\ldots,x_n)=(-x_1,\ldots,-x_n)$. What is the derivative of ...
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1answer
42 views

Kernel of matrix with identity as submatrix

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^k$ be a $C^\infty$ map and let $X=\text{graph}f$, i.e. $$X=\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^k\mid y=f(x)\}.$$ What is the tangent space to $X$ at ...
0
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1answer
42 views

Computing tangent space for quadric

What is the tangent space to the quadric $x_1^2+x_2^2+\ldots+x_{n-1}^2=x_n^2$ at the point $p=(1,0,\ldots,0,1)$? The definition of a tangent space that I know is based on the fact that we have a ...
3
votes
1answer
55 views

Quadric in $n$ dimensions is a manifold?

I can show that the sphere $x_1^2+x_2^2+\ldots+x_n^2=1$ is an $(n-1)$-dimensional manifold by considering the map $f(x_1,\ldots,x_n)=x_1^2+\ldots+x_n^2$, and noticing that $1$ is a regular value of ...
5
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0answers
59 views

Domain invariance for smooth functions

The domain invariance theorem states that for an open set $U\subset \mathbb{R}^n$ and a continuous and injective mapping $f:U\to \mathbb{R}^n,$ the image $f(U)\subset \mathbb{R}^n$ is open. I've read ...
3
votes
2answers
37 views

System of two equations form a manifold

Show that the set of solutions of the system of equations $$x_1^2+\ldots+x_n^2=1$$ and $$x_1+\ldots+x_n=0$$ is an $(n-2)$-dimensional submanifold of $\mathbb{R}^n$. I want to take ...
0
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1answer
41 views

Another exercise from Fleming's Functions of Several Variables.

I'm using Flemming's book Function of Several Variables. In it, the author defines Manifolds like this: Let $1\le r\lt n,\ q\ge1$. A nonempty set $M \subset \mathbb{R}^n$ is a manifold of dimension ...
0
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1answer
51 views

Intersection of a manifold with open set

I'm using Flemming's book Function of Several Variables. In it, the author defines Manifolds like this: Let $1\le r\lt n,\ q\ge1$. A nonempty set $M \subset \mathbb{R}^n$ is a manifold of dimension ...
1
vote
1answer
243 views

How to prove that a vector bundle is trivial iff there are n global sections that form a basis on each fiber?

I can prove the only if part. My attempt to prove if part is the following: Given $n$ global sections $s_1, s_2, ..., s_n$ of a vector bundle $E$ on a smooth manifold $M$ such that they form a basis ...