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

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1answer
43 views

An isomorphism between homology and cohomology with $\mathbb{Z}_2$ coefficients

In the proof of a theorem we did in a class (namely: if $M$ is an odd-dimensional, closed manifold, then $\chi(M)=0$), there's the following step: $$H_k(M;\mathbb{Z}_2)\cong H^k(M;\mathbb{Z}_2)$$ ...
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0answers
45 views

How to construct a diffeomorphism with $p_k \mapsto q_k$?

How to prove the following property? I cannot do anything. Let $M$ be a connected paracompact smooth manifold of dimension $m\geq 2$. Let $(p_k), (q_k)_{k\in \mathbb{N}}$ be sequences on $M$ which ...
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1answer
60 views

If $M$ is a compact manifold what does $\partial M$ mean?

In the generalized form of stokes theorem it states that the integral of the $k+1$ differential form of an operator over a compact manifold $M$ is equivalent to the integral of the $k$ differential ...
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0answers
32 views

A question about the boundary points of manifolds.

A topological $n$-manifold is a second countable Hausdorff space such that every point has a neighbourhood which is homeomorphic to an open ball centred at the origin in $\Bbb{R}^n$. The "centred at ...
4
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1answer
58 views

Looking for proof that $SO(3)$ is a submanifold of $\mathbb R^3$

It seems to be taken for granted in all sources that $SO(3)$ is a submanifold of $\mathbb R^9$. However, the one proof of this that I have been able to find has a step or two that doesn't make alot ...
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1answer
28 views

Example of complete not-connected riemannian manifold

Are there examples of complete Riemannian manifolds which are not connected ? This question follows my previous question. The more I think about it and the less I'm convinced it exists.
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1answer
28 views

A question about manifolds with boundaries.

My topology textbook says the following: Let $S\subset \Bbb{R^2}$ be a closed disc. Then every point in $S$ is contained in a neighbourhood which is homeomorphic to that portion of a ball in ...
2
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1answer
70 views

Riemannian metric and geodesic

For all $p \in \mathbb{R} \smallsetminus \lbrace 0,1 \rbrace$, let $\displaystyle M(p)=\frac{1}{p^{2}(p-1)^{2}}$. Then, let $g_{p} \, : \, (u,v) \, \longmapsto \, uM(p)v$. I am not sure about the ...
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1answer
99 views

Example of Something That's Not A Manifold

Two examples of non-manifolds that I know are the cross and the cone. Also the sphere with a hair isn't a topological manifold. But what's an example of a topological space $X$ such that $X$ is not a ...
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1answer
35 views

Condition for Orientability of Manifold

Let $M^n$, $n>2$ be a manifold and let $f:D\rightarrow M$ be an embedding of the closed $n-$disk in $M$. Prove or Disprove: $M$ orientable iff $M-f(D)$ is orientable. $M$ is orientable iff all ...
2
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1answer
31 views

How can we show a data set satisfies the manifold assumption?

In machine learning, we often assume that a data set lies on a low-dimensional manifold (the manifold assumption), but is there any formal proof saying that assuming the data set satisfies certain ...
2
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1answer
42 views

Quotient of Unit Quaternions by Subgoup (Lie Groups)

Let $G \leq Sp(1)\cong S^3$ (unit quaternions) be a discrete subgroup of order 120 (the Binary Icosahedral Group, not the other one), with presentation $G=<s,t| s^2=t^3=(st)^5>$. $\hspace{2mm}G$ ...
2
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1answer
48 views

Is $\mathbb{R}P^n$ “two-sided” in $\mathbb{R}P^{n+1}$

i.e. Does $\mathbb{R}P^n$ have a tubular neighborhood $N$ such that $N-\mathbb{R}P^n$ is disconnected. My guess is yes, but don't know how to show it convincingly ( or maybe only for $n$ odd, I'm ...
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0answers
39 views

Manifold has uncountable many smooth stuctures if it has one

This is the Problem 1-6 of John Lee's Introduction to smooth manifold: Let $M$ be a nonempty topological manifold of dimension $n\geq1$. If $M$ has a smooth structure, show that it has uncountably ...
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1answer
23 views

Inverse Image of a Regular Value an Orientable Submanifold

Let $f:M^n \rightarrow \mathbb{R}$ be a smooth map, and let $c\in N$ be a regular value. When is $f^{-1}(c)$ an orientable manifold? Note: I know by regular value thm, $f^{-1}(c)$ is a smooth $n-1$ ...
0
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1answer
36 views

Estimate for boundary points and exterior normal vector of bounded domain of class $C^2$

Consider a bounded open set $\Omega\subset\mathbb{R}^d$, s.t. the boundary set $\partial \Omega$ is a manifold of class $C^2$. Let $x,x_0\in\partial\Omega$ be boundary points and $\nu_x$ the exterior ...
3
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1answer
61 views

What's the Differential of this Map $f:S^3\rightarrow \mathbb{R}$

$f:S^3 \rightarrow \mathbb{R}$ is defined as $f(x,y,z,w)=x+zw$, where $S^3= \{(x,y,z,w) | x^2 +y^2 +z^2 +w^2 =1\}$ I tried using a stereographic chart but that got ugly. The function is so simple I ...
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0answers
50 views

Gluing two solid tori along their boundary resulting in a topological manifold

The following question is from a past qualifying exam. Take two solid tori $D^2 \times S^1$, and construct the space $X$ by identifying their boundaries via the map $f \colon \partial D^2 \times S^1 ...
9
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2answers
191 views

Is there a retraction of a non-orientable manifold to its boundary?

It's easy to show using Stokes theorem that a compact orientable manifold with boundary cannot retract to its boundary, by choosing a volume form. But for the non-orientable case I don't know if this ...
3
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0answers
37 views

Second fundamental form of a graph of a function using frame fields

I am trying to recover the second fundamental form $II$ for the graph $\Gamma(f)$ of a smooth real function $f$ in the Euclidean space $\mathbb{R}^3$ using first order frame fields. I have worked with ...
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1answer
31 views

Difference betwee parameterization and embedding of manifolds

What is the difference between embedding and parameterization? Why, for example, we say Gauss parameterization of a convex hypersurfaces, and we don't call it an embedding?
7
votes
2answers
120 views

Smooth surfaces that isn't the zero-set of $f(x,y,z)$

The zero-set of any smooth function $f(x,y,z)$ with a non-vanishing gradient is a smooth surface. I was wondering if the reverse is true: is every smooth surface in $E^3$ the zero-set of some smooth ...
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3answers
68 views

May directed graph be embedded into manifold?

May directed graph be embedded into manifold?How ?and what is the condition?
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1answer
29 views

A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
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1answer
13 views

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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0answers
64 views

Submanifold of $2\times 2$ Complex Matrices Using Transversality?

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
46 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 ...
4
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2answers
49 views

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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2answers
96 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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2answers
60 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 ...
0
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1answer
75 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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0answers
32 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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0answers
15 views

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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0answers
45 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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0answers
63 views

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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0answers
47 views

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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0answers
43 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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1answer
79 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 ...
3
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1answer
62 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
27 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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0answers
57 views

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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0answers
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 ...
2
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1answer
34 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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0answers
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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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0answers
35 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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0answers
41 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. ...
4
votes
1answer
81 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 $$\int: \Omega_c^n(\Bbb R^n)\to\Bbb R$$ $$\alpha(x)dx^1\land...\land dx^n\mapsto \int_{\Bbb R^n}\alpha(x)dx^1\land...\land dx^n$$ ...
3
votes
2answers
72 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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0answers
15 views

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 ...