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

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11
votes
1answer
549 views

Problem 3-38 in Spivak´s Calculus on Manifolds

This is not homework. Problem 3-38 reads: Let $A_{n}$ be a closed set contained in $(n,n+1)$. Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for ...
3
votes
2answers
449 views

Green's Function for Operator

I'm trying to show that the Green's function for the Laplace operator $-\nabla^2$ is badly behaved at infinity. I.e. $$\int d^dx|G(x,y)|^2=\infty$$ for $d=1,2,3$. What happens when $d>4$? I know ...
4
votes
2answers
899 views

The graph of a smooth real function is a submanifold

Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ which is smooth, show that $$\operatorname{graph}(f) = \{(x,f(x)) \in \mathbb{R}^{n+m} : x \in \mathbb{R}^n\}$$ is a smooth submanifold of ...
15
votes
1answer
355 views

Is there any holomorphic version of the tubular neighborhood theorem?

This question arised when I was studying Beauville's book 'Complex Algebraic Surfaces'. Castelnuovo's theorem says that a smooth rational curve $E$ on an algebraic surface $S$ is an exceptional ...
0
votes
1answer
36 views

How do I show this integral surface area relationship?

Let alpha=xdy-ydx and let M be a compact domain in the plane R^2. Show that the integral along the boundary of M of alpha is twice the surface area of M.
3
votes
1answer
204 views

Levi-Civita connection

Well if $\Sigma$ is a submanifold of $R^{n+p}$ and $\{e_i,e_\alpha\}$ is orthonormal frame over $\Sigma$ where the $e_i$'s are tangent and the $e_\alpha$'s are normal to $\Sigma$. Can anyone prove ...
6
votes
1answer
146 views

Looking for an atlas with 1 chart

Can we provide the set $\{(x,y,z)\in\mathbb{R^3}|x^2+y^2=1\}$ with a 2-dimensional manifold structure involving only 1 chart? I can see it with 2 charts with cylindrical coordinates, but not with only ...
1
vote
1answer
79 views

Virtual knot diagrams on surfaces with genus?

To the best of my limited understanding, a virtual knot diagram may be thought of as the projection of an embedding of $\mathbb{S}^1$ in a 2-manifold with genus onto $\mathbb{R}^2$. That is to say it ...
4
votes
2answers
174 views

Connected manifold + hole = connected manifold?

If I have a connected manifold and I poke a hole in the interior of the manifold, it seems obvious that the manifold is still connected. But how would you prove this? More precisely, let $M$ be a ...
5
votes
1answer
161 views

special covering of a non-compact manifold

I'm very stuck on the following exercise in the book "A Comprehensive Introduction to Differential Geometry V.1" by Michael Spivak: Let $M^m$ be a smooth connected non-compact manifold. Show that $M$ ...
2
votes
1answer
195 views

One-point compactification of manifold

A question from Introduction to Topological Manifolds: 4-28. Suppose $M$ is a noncompact manifold of dimension $n \ge 1$. Show that its one-point compactification is an $n$-manifold if and only if ...
1
vote
1answer
60 views

How can I show that $\det(v_1,v_2,\ldots,v_n)=dx_1\, dx_2\cdots dx_n(v_1,v_2,\ldots,v_n)$?

I wanted to use the definition of a wedge product which says $λ_1λ_2\cdots λ_k(v_1,v_2,\ldots,v_k)=\det(λ_i(v_j))$ with $1<i,j<k$ but I'm not sure if that even can work
1
vote
0answers
30 views

Coordinate patch of an n-sphere? [duplicate]

Possible Duplicate: Analogue of spherical coordinates in $n$-dimensions If we take a 2-sphere of radius a, we can define $ f(z, t) = (\sqrt{a^2-z^2}\cos t,\sqrt{a^2-z^2}\sin t, z) $ it's a ...
2
votes
1answer
98 views

Ramified coverings of a manifold

Maybe someone could help me with a bit of alebraic topology. Take $M$ a $n$-manifold with $n \geq 3$ , and $V$ a submanifold of codimension $2$ in $M$. Assume $H_{n-2}(V) = 0$. I've read that under ...
1
vote
1answer
110 views

Homology of a $3$-manifold obtained by rational surgery on an $m$-component link

I was trying to understand homology of a $3$-manifold $M$ obtained by rational surgery on an $m$-component oriented link $L$. I have a few questions regarding the following paragraph in the book ...
3
votes
2answers
522 views

Integral over a torus

I've been asked to solve the following integral: $$\int_{\mathbb{T}^2} xyz \, dw\wedge dy$$ where $\mathbb{T}^2\subset\mathbb{R}^4$ is the 2-torus defined by: $$w^2+x^2=y^2+z^2=1$$ I've tried ...
1
vote
1answer
220 views

Prove that $[0,\infty)$ is not a manifold.

Prove that $[0,\infty)$ is not a manifold. Using diffeomorphisms and the implicit function theorem perhaps.
1
vote
3answers
224 views

Express the volume of an n-sphere in terms of the volume of an n-1 dimensional ball

It's an exercise from Munkres "Analysis on Manifolds" Chapter 5, "Integrating a scalar function over a manifold". Due to the suggestion, I'm repeating the question here: Express the volume of an ...
10
votes
2answers
1k views

Is the Nash Embedding Theorem a special case of the Whitney Embedding Theorem?

The Whitney Embedding Theorem states that every smooth manifold can be embedded in Euclidean space. The Nash Embedding Theorem states that every Riemannian manifold can be embedded in Euclidean ...
3
votes
2answers
902 views

Does the Implicit mapping theorem imply the inverse mapping theorem?

Does the Implicit mapping theorem imply the inverse mapping theorem?
1
vote
2answers
79 views

About regular surfaces

I never had seen this exercise, but I'm confused again, I don't know what I have to use. I have the surface $S=\{(x,y,z)\in \mathbb{R}^3|xy+xz+yz=1,x>0,y>0,z>0\}$, is $S$ regular?. Then, if ...
4
votes
2answers
562 views

Showing something isn't a manifold

So I'm following some notes that are introducing manifolds with pretty minimal prerequisites. What I want to do is show where the image of $\phi: \mathbb{R}\rightarrow \mathbb{R^2}$ $t\mapsto ...
2
votes
1answer
396 views

Book recomendations for Smooth manifolds.

I want to learn about smooth manifolds, I have never studied them before, but I have a good background in Algebra. Can any one recomend some good introductory books? Thanks
1
vote
1answer
99 views

Zero Locus of Functions is a Submanifold

Suppose $f_1,\dots, f_d$ are a set of real-valued functions on a smooth manifold $M$. Let $N$ be the zero locus of the $f_i$. Suppose the $\textrm{d}f_i$ span a subspace of the cotangent space of $M$ ...
2
votes
2answers
135 views

Deciding whether a given set is a manifold

This question should be extremely elementary but it stumped me somewhat. I know the definition of a (topological/smooth) manifold but I seem to have trouble when it comes to deciding whether a given ...
1
vote
2answers
319 views

Extending a Set of Linearly Independent Vector Fields to a Basis

My question is this. Suppose we are given some smooth vector fields $X_1, X_2,..., X_k$ which are linearly independent at all points in a neighborhood $U$ (EDIT: diffeomorphic to a ball) of $R^n$. Do ...
0
votes
0answers
47 views

Problem with a proof of one of characterization of manifold in $\mathbb R^n$

Let $M \subset \mathbb R^n$, $k \in \mathbb N$. Assume that $M$ is a $k$-dimensional manifold in $\mathbb R^n$, i.e. for each $x \in M$ there exists an open set $W \subset \mathbb R^k$ and a smooth ...
1
vote
1answer
242 views

How to find the frechet derivative at $A\rightarrow A^{-1}$ mapping?

I am reading on my own the Lectures on the Geometry of Manifolds (http://nd.edu/~lnicolae/Lectures.pdf ) , and got stuck in solving the exercise 1.1.3 (b) . The 1.1.3 (b) is : Let F: $U\rightarrow ...
1
vote
1answer
104 views

Quotient of $R $ by $2πZ$

Let the additive group $2πZ$ act on $R $ on the right by $x · 2πn = x +2πn$, where $n$ is an integer. Show that the orbit space $R/2πZ$ is a smooth manifold.
1
vote
0answers
66 views

show that subset of complex projective space is a submanifold

Let n, m $\in \mathbb{N}$. I'm trying to show that $M(n,m) = \{[z_0 : z_1 : … : z_n] \in \mathbb{C}P^n | \sum^{n}_{i=0} z^m_i = 0\}$ is a submanifold and codim(M(n,m))=2. My idea was to use ...
2
votes
0answers
61 views

Complex structure on the product of two complex Kaehler manifolds

I have the following question: Let $(M,J_{M}, \omega_{M})$ and $(N,J_{N}, \omega_{N})$ be two Kaehler manifolds. I consider $M \times N$. Can I introduce on $M \times N$ a complex structure ? I think ...
9
votes
1answer
512 views

intrinsic proof that the grassmannian is a manifold

I was trying to prove that the grassmannian is a manifold without picking bases, is that possible? Here's what I've got, let's start from projective space. Take $V$ a vector space of dimension n, and ...
0
votes
0answers
33 views

Why $C^{\infty}(M)$ module of sections of a vector bundle $E\rightarrow M$ is a reflexive module?

As the title saying, Why $C^{\infty}(M)$ module of sections of a vector bundle $E\rightarrow M$ is a reflexive module? Here we are considering vector bundles with finite-dimensional fibers.
3
votes
1answer
138 views

O(n) as embedded submanifold

I want to show that the set of orthogonal matrices, $O(n) = \{A \in M_{n \times n} | A^tA=Id\}$, is an embedded submanifold of the set of all $n \times n$ matrices $M_{n \times n}$. So far, I have ...
0
votes
1answer
300 views

Universal cover of complete hyperbolic surfaces and torsion-free, discrete groups of isometries of $\mathbb{H}^2$

I'm taking a course this semester, and in it we proved that any complete hyperbolic surface is universally covered by $\mathbb{H}^2$. The text, found at ...
0
votes
1answer
200 views

cotangent bundle splits as a product?

Let $M$ and $N$ be two Riemannian manifolds with Riemannian metrics $g$, $h$ respectively. We consider the product $M \times N$ with metric $g \oplus h$. By the metric we get an isomorphism of bundles ...
3
votes
1answer
380 views

$S^1$ is the only compact connected 1-manifold.

I want to find a proof of $S^1$ is the only compact connected 1-manifold. (In here, manifold means Hausdorff, locally euclidean space) Is there any reference? Or is it easy and can be proved simply?
0
votes
2answers
103 views

Tangent space on the north pole of $S^2$

We have 3 different (but equivalent) definition of tangent space, one of them is by equivalent class of smooth path, i.e. let $c:(-1,1)\rightarrow M$ be a path on $M$ with $c(0)=p$, then $c_1\sim c_2$ ...
2
votes
1answer
132 views

Second Fundamental form in terms of defining function

I have an m-dimensional riemannian manifold M and an n-dimensional submanifold N that is given by $N = f^{-1}(0)$, where $f: M \longrightarrow \mathbb{R}^{m-n}$ ($0$ is supposed to be a regular value ...
0
votes
1answer
162 views

The smoothness of an inclusion map

Let $M$ and $N$ be manifolds and let $q_0$ be a point in $N$. Prove that the inclusion map $i_{q_0} : M \to M×N : p \mapsto (p,q_0)$, is $C^\infty$.
4
votes
2answers
496 views

Any example of manifold without global trivialization of tangent bundle

It is said for most manifolds, there does not exist a global trivialization of the tangent bundle. I am not quite clear about it. The tangent bundle is defined as $$TM=\bigsqcup_{p\in M}T_PM$$ So is ...
1
vote
0answers
75 views

Problems about dual map, cotangent bundle.

I have no idea what dual space and dual map are, so have much trouble understanding cotangent bundle when reading Lee's book. First of all, can anyone give me a introduction what the dual map and ...
6
votes
2answers
121 views

How does smoothness prevent “singularities”?

This is a refinement of one of my earlier questions (I failed to put into words what I really wanted to ask). First of all, I'm not sure "singularity" is the correct word to use hence the quotes. ...
6
votes
1answer
441 views

Geodesics on the torus

[This is a follow-up to my question Is there a Möbius torus?] Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus: There are five ...
4
votes
1answer
532 views

How to prove figure eight is not a manifold? [duplicate]

Possible Duplicate: A wedge sum of circles without the gluing point is not path connected I know that figure eight is not a manifold because its center has no neighborhood homeomorphic to ...
1
vote
0answers
292 views

Extension of a smooth function on a set of manifold

I encountered the following proposition: If a function is smooth on an arbitrary set $S\in M$, where $M$ is a smooth manifold, then it has a smooth extension to an open set containing $S$. It seems ...
1
vote
0answers
56 views

Can you consider a directed graph the discretization of a 2-manifold equipped with a vector field?

Can you consider a directed graph the discretization of a 2-manifold equipped with a vector field? Thanks
0
votes
1answer
385 views

How can I find the winding number of a curve?

I need to find the winding number of the closed curve $c(t)=(a \cos(t),b \sin(t))^T $, where $a,b > 0$ and $c:[0,2\pi) \to\mathbb{R}^2\setminus\{0\}$. I don't understand how to do this.
6
votes
2answers
756 views

Manifold with different differential structure but diffeomorphic

I'm new to differential geometry and reading Lee's book Manifold and Differential Geometry. In the first chapter, he mentioned the following two maps on $\mathbb{R}^n$: (1) $id: (x_1,x_2\cdots x_n) ...
0
votes
1answer
225 views

The necessary and sufficient condition for diffeomorphism

I came to the following proposition: Let $M$ and $N$ be two smooth manifolds with respective maximal atlases $\mathcal{A}_M,\mathcal{A}_N$. Then a bijection $f:M \rightarrow N$ is a diffeomorphism if ...