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

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2answers
171 views

Why do differential geometry textbooks bother with equivalence classes of smooth structures?

In contemporary textbooks on differential geometry, the definition of smooth manifolds is given in a (IMHO) awkwardly obfuscated way, by saying that a smooth manifold is a topological space endowed ...
1
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1answer
60 views

Manifold in $\mathbb{R}^2$

Is the set $\left\lbrace (x,y) \in \mathbb{R}^2 \mid x^2 = y^2 \right\rbrace $ a manifold in $\mathbb{R}^2 $? I know I could use the level set theorem if I had $\left\lbrace x \in \mathbb{R}^2 ...
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2answers
83 views

Manifolds and Topological Spaces

from my understanding of manifolds they are structures defined on topological spaces. So if M is a manifold defined on a topological space $(X,\tau)$ and $X\subseteq\mathbb R^3$, does this mean $M$ is ...
0
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1answer
71 views

Is this a topological manifold?

Consider the map $w \rightarrow (w^3, w^2)$ from $\mathbb{C}$ to $ \mathbb{C}^2$. Is the image of this map a topological manifold? I think the map is bijective and continuous, but is not a ...
2
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2answers
112 views

Homologous surfaces in three-manifolds

Let M be a 3-manifold. Let $S$ and $T$ be properly embedded surfaces in $M$ such that $[S] = [T] \in H_2(M, N(\partial S)) $. Is it true that we can isotope $\partial S$ so that it coincides with ...
1
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1answer
205 views

Does a diffeomorphism always map boundaries to boundaries?

If $F:\overline{M} \to \overline{M'}$ is a diffeomorphism between two open bounded domains $M$ and $M'$ in $\mathbb{R}^n$, what conditions do I need on $F$ or $M$ or $M'$ to make sure that $F(\partial ...
0
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1answer
43 views

Assumptions required for an implicitely defined surface/manifold to have a specified dimension

What are some normal assumptions made on implicitly defined manifolds? More specifically, by implicitly defined manifold, I mean the definition of a surface such as $g(x,y)=x^2+y^2-1=0$ for the ...
0
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1answer
153 views

$C^1$ domains vs. Lipschitz domains in PDEs, do we need transformation of coordinates?

I am getting a bit confused. In the definition of $C^1$ manifold in Renardy and Rogers, they say that $\partial\Omega$ is of class $C^1$ if every point on $\partial\Omega$ has a neighbourhood within ...
3
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0answers
84 views

Does a “typical” reproducing kernel on a manifold generate an infinite-dimensional RKHS?

Let $M$ be a finite dimensional manifold and $k:M\times M\rightarrow\mathbb{R}$ a smooth reproducing kernel. Is the set of all such $k$ with the property that the reproducing kernel Hilbert space ...
2
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1answer
57 views

Question on Construction in Spivak's *Calculus on Manifolds*, induced transformations

First I quote the relevant passage (page 89): If we consider now a differentiable function $f : \mathbb R^n \to \mathbb R^m$ we have a linear transformation $Df(p): \mathbb R^n \to \mathbb R^m$. ...
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1answer
59 views

The set $\{(x,y) \in \mathbb{R}^2 | x^3=y^2 \}$ is not a submanifold

Prove that the set $S=\{(x,y) \in \mathbb{R}^2 | x^3=y^2 \}$ is not a submanifold. This is the exercise from the book and I cannot understand why the chart $\phi :S \rightarrow \mathbb{R}$, ...
4
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2answers
241 views

Most important aspects of differential geometry for general relativity

I'm an undergraduate getting ready to take a graduate course in general relativity next quarter. I purchased Wald's General Relativity (who incidentally will be teaching the class) in order to get a ...
4
votes
1answer
112 views

Concrete non trivial computation of Morse homology

I am studying Morse homology and have found only examples on spheres and tori so far. Of course the homology of these manifolds is better understood by other more standard methods, so I am having ...
0
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0answers
48 views

Some questions on definition of Grassman manifolds

In W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry , Page 64. We use coordinate correspondences $\phi_j : U_j \to R^{k(n-k)}$ in order to define Grassman manifolds ...
4
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1answer
85 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
101 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
61 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
49 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
97 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 ...
0
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0answers
54 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 ...
2
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1answer
124 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 ...
2
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0answers
29 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
75 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
votes
1answer
107 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
122 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 ...
4
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0answers
267 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 ...
0
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0answers
48 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 ...
2
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0answers
47 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, ...
0
votes
1answer
25 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
votes
1answer
137 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
77 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
votes
2answers
87 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
62 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
144 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
111 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
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1answer
143 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
24 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 ...
1
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1answer
56 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
51 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
120 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
30 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
69 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 ...
1
vote
2answers
133 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. ...
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1answer
121 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
279 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 ...
4
votes
2answers
86 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
votes
1answer
49 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
46 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
76 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
45 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 ...