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

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Reference for transformation of integrals over Lipschitz boundaries

Let $D\subseteq\mathbb{R}^d$, $d\ge 2$, be a bounded Lipschitz domain. Then according to page 314 of [Function spaces, Alois Kufner, 1977] one can define a surface integral of a real valued function ...
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42 views

Submanifold of $2\times 2$ complex matrices?

Let $M$ be the set of all $2\times 2$ complex matrices with $a_{21}=\bar a_{12}$. $M$ is a smooth manifold diffeomorphic to $\mathbb{R}^6$. Let $W$ be the subset consisting of all matrices in $M$ with ...
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1answer
25 views

extension of a local orthonormal frame on a hypersurface

Let $N$ be a $(n+1)$-dimensional Riemannian manifold and $M\subset N$ a Riemannian hypersurface (embedded or immersed). Let $M$ and $N$ be oriented and choose a unit normal vector field $\nu$ along ...
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Geometrically, what is the stereographic projection of a closed $n$-ball?

To show $\overline{B^n}$ is a $n$-manifold with boundary, apparently there is a trick to use stereographic projection after subtracting out the radius connecting $0$ to the north pole. I'm familiar ...
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67 views

Cohomology Ring of Klein Bottle over $\mathbb{Z}_2$

I am trying to show that the cohomology ring of the Klein bottle with $\mathbb{Z}_2$ coefficients is $H^*(K,\mathbb{Z}_2) \cong \mathbb{Z}_2[x,y]/(x^3,y^2, x^2y)$. What I know: ...
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55 views

Closed 4-manifolds have CW-complex?

It looks like for compact 4-manifolds this question is open: When is a compact topological 4-manifold a CW complex? How about if we just consider closed 4-manifolds, does that have an answer/make the ...
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1answer
52 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 ...
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30 views

Why are compact and noncompact manifolds without boundary called closed manifolds and open manifolds, respectively?

Why not just call them compact and noncompact manifolds? Isn't the general assumption that manifolds have empty boundary unless stated otherwise?
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Definition of Non-characteristic Manifold

What is the definition of non-characteristic manifold in the following context. "Since $ f (x, y)=0 $ is assumed to be a non-characteristic singularity manifold, we have $ f_{x}\neq 0 $." Thanks, ...
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41 views

An Atlas for $\mathbb{R}/{2\pi \mathbb{Z}} $

I've been having some difficulty finding an atlas for $\mathbb{R}/{2\pi \mathbb{Z}}$. The way I have been thinking of this so far is by using the standard projection map $\pi$ on open intervals of ...
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Role of the Thom space in the Pontryagin-Thom construction

I am trying to understand the Pontryagin-Thom theorem; especially how the Thom space comes into play. Just to bring everyone on the same page: I am specifically talking about the construction of an ...
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Surgery and Euler Characteristic

I am trying to find out how a $(p,n-p)$ surgery affects the Euler Characteristic of an orientable, $n-$ dimensional, compact manifold. Call the initial manifold $M$ and the post-op manifold $M'$. This ...
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41 views

manifold structure on a finite dimensional real vector space

I'm reading Warner's Foundation of Differential Manifolds and Lie Groups. I don't get how the finite dimensional real vector space gives a natural manifold structure. Someone has asked it before ( ...
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64 views

why some solid can have no surface? [closed]

For solid construction, I can understand the closed surface has no edges. But i cannot understand why some solid can have no surface (except just lines?), any other solid which can have no surface?
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1answer
71 views

Does a compact manifold require non zero Ricci curvature?

Imagine we have a Riemanian compact manifold. Does the compactness necesarily make its curvature non zero? If the answer were no, does anyone know any such manifold with isometry group ...
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1answer
54 views

Differential Forms on submanifolds

Say I take an embedded submanifold of $\Bbb R^n$, like the sphere. Any differential form on $\Bbb R^n$ can be restricted to the sphere. My question is this: is any differential form on the sphere (or ...
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1answer
26 views

On Stein manifolds and constant functions

Stein manifolds are defined here: http://en.wikipedia.org/wiki/Stein_manifold#Definition Obviously, M is Stein implies that there is a non-constant holomorphic function defined in it. Is the converse ...
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40 views

Criteria for being a submanifold

Let $N$ be a subset of a smooth manifold $M$. I want to proof (or a reference for) the following criteria: Every connected component of $N$ is a locally closed submanifold of $M$ if (and only if?) ...
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A 1-form on a smooth manifold is exact if and only if it integrates to zero on every closed curve

I am stuck on the following problem, which comes from a old qualifying exam. Prove that a 1-form $\phi$ on $M$ is exact if and only if for every closed curve c, $\int_{c} \phi =0$. One way is an ...
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33 views

Cellular structure on a manifold [duplicate]

Is it always possible to put a cell structure on a manifold? in other words is it possible to decompose a manifold as a CW complex? I know that by Morse theory we always have a handle decomposition of ...
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1answer
31 views

A diffeomorphism between manifolds (or surfaces) that preserves the mean value of functions

Let $M$ and $N$ be two Riemannian manifolds with $f:N \to M$ a diffeomorphism with the following properties: for all $u \in H^1(M)$, $\hat u := u\circ f$ satisfies $\hat u\in H^1(N)$ and furthermore ...
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Integration on Manifold

I am beginning my studies on integration on manifolds and i have some theorical questions. First, in all books that I saw they says that the singular p - simplex (or p - cube) are continuous mapping ...
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33 views

Question about a particular estimate in Riemannian geometry.

I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...
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39 views

Tangent bundle is orientable

I am having some trouble finishing a proof that the tangent bundle of any manifold is orientable. What I've done so far is calculate the transition function between two standard charts on the bundle. ...
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1answer
57 views

Orientation of hypersurface

Some books on mean curvature flow (e.g. Mantegazza, Ecker) state that an embedded hypersurface in $\mathbb{R}^{k+1}$ is orientable (Mantegazza page 3, Ecker page 110). In other words, they assume the ...
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1answer
21 views

Question on exact $n$ form with compact support

I've encountered with the following problem: Consider the map $$\int: \Omega_c^n(\Bbb R^n)\to\Bbb R$$ $$\alpha(x)dx^1\land...\land dx^n\mapsto \int_{\Bbb R^n}\alpha(x)dx^1\land...\land dx^n$$ ...
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61 views

A proper local diffeomorphism between manifolds is a covering map.

The following is an exercise taken from "Manifolds and Differenial Geometry" by Jeffrey M. Lee. Let $\widetilde M$ and M be (connected) $C^r$ manifolds. Let $f: \widetilde M \to M$ be a proper map ...
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Why $\dim(K\cap N(I)) +1 \geq \dim(K)$ if the index of the index form equals 1?

Let $c:[0,1]\rightarrow O$ a geodesic, $O$ Riemann manifold and let $\mathcal{W}$ the space of piecewise smoothl normal vector fields $W(t)$ along $c$ mit $W(0)=W(1)=0$. $N(I)$ is the nullspace of ...
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1answer
50 views

Wedge Sum of Two Spheres Homotopy Equivalent to a Compact Manifold?

Let $X=S^2$v $S^2$ (wedge sum). The homology groups are $H_0(X,\mathbb{Z})= \mathbb{Z}$, $H_1(X,\mathbb{Z})= 0$, and $H_2(X,\mathbb{Z})= \mathbb{Z} \oplus\mathbb{Z}$. I can see that $X$ is not ...
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1answer
37 views

Is the composition of a homeomorphism with itself orientation-preserving?

Just a short question about the degree of a homeomorphism. So, I understand that in the continuous setting we define the degree of a map $\ f: M \rightarrow M$ on a connected orientable manifold as ...
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1answer
60 views

Deleting a subset from a topological manifold

Consider an $n$-topological manifold $M$. We remove a subset $A$ from $M$. Are there cases where $M-A$ is no longer a topological manifold. In case we suppose that $M-A$ is still a manifold, what ...
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1answer
74 views

Classical tensor analysis and Tensors on Manifolds

I learned tensors the bad way (Cartesian first, then curvilinear coordinate systems assuming a Euclidean background) and realize that I am in very bad shape trying to (finally) learn tensors on ...
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2answers
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Some questions about the proof of the General Linear Group being a manifold.

I understand the idea behind proving that GL(n,$\mathbb{R}$) is a smooth manifold by first using the fact that it is isomorphic to $\mathbb{R}^{n^{2}}$ and using the continuity of the determinant ...
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1answer
63 views

Signature of $S^2 \times D^2$

Every closed connected oriented $4$-manifold has a signature, defined via a cohomological intersection form. In Turaev's book Quantum Invariants of Knots and 3-Manifolds the definition of a certain ...
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1answer
51 views

Confused about the possibility of different differentiable structures.

In Loring Tu's "An Introduction to Manifolds" an atlas on a manifold is a collection of coordinate charts that are pairwise compatible and cover the manifold. A smooth manifold is defined to be a ...
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31 views

Submersions define Foliations

Let $M$ be a $C^\infty$ manifold of dimension $m$. A $C^r$ foliations of dimension $n$ of $M$ is a $C^r$ atlas $\mathcal{F}$ of $M$ which is maximal (not needed) with the following properties: a) If ...
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1answer
45 views

Triangulations, PL-triangulations and related conecpts

I'm confused about various definitions of triangulations and piece-wise linearity. I read, for example, on wikipedia "..the question of whether all topological manifolds have triangulations is an ...
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1answer
53 views

The graph of $x\mapsto |x|$ cannot be the image of an immersion.

How can one prove that the set $\{(x,|x|)\in \mathbb{R}^2 \mid x\in \mathbb{R}\}$ cannot be the image of an immersion of a smooth manifold? This was my homework exercise in a course about ...
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1answer
53 views

General introduction to orbifolds?

Where should I go to learn about orbifolds? I am interested in a general introduction that gives precise definitions and clear explanations. I have a fair background in topological and smooth ...
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2answers
41 views

How many 2-manifolds can be covered by a single chart?

Every 1-manifold is either homeomorphic to a sphere or to the real line. Therefore, one can trivially say that all 1-manifolds (without boundary) covered by a single chart are equivalent to ...
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28 views

Does a metric has a neighboorhood in which signature is the same?

suppose $M$ is a manifold and $g$ is a metric with a specific signature on it. there are at least 2 topologies for metrics that I am aware of. Fine $C^k$ topologies and the topology of metrics as ...
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1answer
65 views

Atlas/chart for a Hyperboloid

given is the following hyperboloid: $$H = \{(x,y,z) \mid \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\},$$ where a,b,c are free parameters. I have to find an $C^\infty$-atlas for H. In ...
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1answer
18 views

natural projection on a slice

I'm currently studying Warner's book "Foundations of Differentiable Manifolds and Lie Groups". Within the proof of the Frobenius Theorem he is constructing a slice $S$ of a coordinate system ...
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1answer
46 views

A question about the condition of Frobenius theorem

I puzzled about the condition of Frobenius theorem: Condition FR1: Let $X$ be a manifold, $E$ is a subbundle of $TX$,vector fields $ ξ,η $ lie in $E$(i.e. $ ξ(x),η(x)\in E_x $),then bracket $[ξ,η]$ ...
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What is a manifold with cusp?

What I am primarily currently learning about is hyperbolic geometry and methods to find hyperbolic structures on triangulated manifolds. I see phrases such as 'cusp ends' and 'manifold with one cusp' ...
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1answer
48 views

A question about tangent subbundle

Let $x$ be a manifold, $E$ is a subbundle of $TX$ , my question is : Can you give example such that vector fields $\xi ,\eta$ lie in $E$,but bracket $[\xi ,\eta]$ does not lie in $E$ in some point ...
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1answer
45 views

An almost complex structure on real 2-dimensional manifold

Why an almost complex structure on real 2-dimensional manifold is integrable?
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Why are homotopy spheres spin-cobordant in dimensions divisible by 4?

Manifolds are assumed to be smooth having dimensions $\geq5$. As usual, $\Omega^{Spin}_{*}$ denotes the spin bordism ring and $\Omega^{SO}_{*}$ the oriented bordism ring. Let $\Sigma^n$ be a closed ...
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59 views

Calculate Geodesic Path of $N\times N$ matrix on Riemannian manifold of fixed rank

If I have two matrices $A(0)$ at $t=0$ and $A(1)$ at $t=1$, they are $N\times N$ matrices, and they are on the Riemannian manifold of rank $K$. How to calculate the geodesic path $A(t)$? I haven't ...
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Decomposition of Laplacian into tangental and normal components w.r.t. submanifold

If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the ...