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Questions tagged [compact-manifolds]

For questions regarding the structure and properties of compact manifolds.

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1answer
35 views

Stiefel Whitney class and intersection form

Why is the second Stiefel-Whitney Class of a closed oriented 4-manfifold, $M^{4}$, a characteristic element for its intersection form? Precisely, why must the following identity hold for closed ...
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2answers
72 views

Why is this 3-manifold irreducible?

Let $M = \mathbb{R}\mathbb{P}^2 \times \mathbb{S}^1$. It is a prime 3-manifold, but it cannot be reducible, since the only reducible prime connected 3-manifolds are the $\mathbb{S}^2$-bundles over $\...
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0answers
85 views

Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?

Let $M$ be a compact connected manifold-with-boundary such that $\circ M \neq \emptyset$, where $\circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $\circ ...
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0answers
24 views

Minimizing elastic energy of shrinking balls

I am new to variational analysis and I am currently working in the following setup: We denote the $2$-sphere with radius $r>0$ by $S_r^2$ and $S^2:=S_1^2$. In coordinates $(\theta,\varphi)$, ...
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1answer
19 views

Existence of an open set such that the map is submersive

Let $f:M\to N$ be a smooth surjective map between two compact manifolds. Is there always an open set $U\subset M$, s.t. on $U$, $f$ is submersive?
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1answer
20 views

Is there Method to visualize the object $Disc \times Disc$?

For me its clear how to build up the object $S^1 \times S^1$ , who is our old friend torus, but the product of 'interior'of these objetcs doesn't look clear to me how to build up I tried 'forget' ...
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0answers
28 views

Extension of embedding from a compact submanifold into $\mathbb R^{n}$

Suppose $M$ is a manifold, $N$ is a compact submanifold(with non-empty boundary) of same dimension, $f:N\rightarrow\mathbb R^{n}$ is an embedding into sufficiently-high-dimensional Euclidean space. ...
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1answer
55 views

Why does any connected closed $m$-manifold that can be embedded in $E^{m+1}$ bounds a compact connected $(m+1)$-manifold?

I am reading Sheila Carter and S.A. Robertson's paper Relations Between a Manifold and its Focal Set. In this paper, they use the following facts: Any closed $m$-manifold $M$ that can be embedded ...
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2answers
36 views

Question about critical point of function on compact manifold

How can I deduce that $f$ only has finitely many non-degenerate critical points by this function only has non-degenerate critical point? And how can I use the compact manifold’s properties to solve ...
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1answer
60 views

Existence of a special homeomorphism on $\mathbb{T}^2$.

Let $A, B$ be closed topological subspaces of $\mathbb{T}^2$. Suppose that $A$ and $B$ are homeomorphic as topological spaces. My Question: Is it possible to construct a homeomorphism $h: \mathbb{...
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0answers
10 views

Would this subset dense in 2-dim compact manifold?

Let $X$ be a $2$-dimensional riemannian manifold. Let $x_0\in X$ be a point. Let $x_1$ be a point in $X$ such that $d(x_1,x_0)$ is the supremum of $d(x,x_0)$ where $x\in X$. If $x_n$ is well-defined, ...
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1answer
41 views

distance function in context of elastic energy of non euclidean thin bodies

I am currently reading a paper about elastic energy of non euclidean thin bodies, about which I might want to write my thesis for my bachelors degree. You can find it here: https://arxiv.org/abs/1801....
3
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1answer
60 views

Showing algebraic dependence of meromorphic functions on a compact Riemann surface

I have been given the following question to do: Let $f,g$ be meromorphic functions on a compact Riemann Surface $R$. Show that there is some polynomial such that $P(f,g) = 0$ (i.e. show that any two ...
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1answer
63 views

Embedding a compact manifold in $\mathbb{R}^N$

I have an attempt at solving the following problem. This is not so much a question asking for a solution in general, but more on how to complete my own. Let $M^n$ be a compact smooth manifold. Show ...
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1answer
84 views

Smooth no-where vanishing form

Does there exist any no-where vanishing smooth $1$-form on $S^2$. I , think there is such one. For example, consider the smooth $1$-form $\omega=dx+dy+xdz$ on $\Bbb R^3$ consider the pull-back of $\...
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1answer
48 views

Are non-orientable manifolds necessarily compact?

If not, what is an example of a non-compact, open manifold that is non-orientable? So if non-orientability $\Rightarrow$ compactness, is there a theorem and what is the proof?
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1answer
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Retract of Compact $2$-manifold

Consider by $S_g := T \# T \# ... \# T$ the $g$ times connected sum of tori $T$. Obviously since it is a compact $2$-manifold in light of the famous classification of compact $2$-manifolds $S_g$ is ...
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0answers
33 views

Blow-up and connected sum

I'm trying to understand the proof (in Complex Geometry-Daniel Huybrechts ) that the Blow-up at a point $p$ of a complex manifold $X$ of dimension $n$, denote it by $B\ell_p(X)$ is diffeomorphic to $X\...
3
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1answer
51 views

Existence of boundary cylindrical neighborhood for a compact manifold

Let $M$ be a compact riemannian manifold with boundary $\partial M\neq \varnothing$. I would like to show that there is some neighborhood $U$ of $\partial M$ which is diffeomorphic to $[0,a)\times \...
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0answers
33 views

Reference request Eigenspace decomposition Hodge Laplacian on forms on manifolds with boundary

I know that on a connected, compact, oriented Riemannian manifold without boundary the Hodge Laplacian $\Delta_k=(d+\delta)^2$ (acting on $k$-forms) admits an orthogonal eigenspace decomomposition of $...
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1answer
203 views

Homology Groups of Non Orientable Manifold

Let $M$ be a compact, connected $n$-manifold. Consider the homology groups $H_n(M)$ with coefficients in $\mathbb{Z}$. It is well known that if $M$ is not $\mathbb{Z}$-orientable, then we have $H_n(M)...
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2answers
62 views

Show that if $A\subseteq \mathbb R^n $ is closed and bounded(compact)

Show that if $A\subseteq \mathbb R^n $ is closed and bounded(compact), then $A\times A \subseteq \mathbb R^n\times \mathbb R^n $ is closed and bounded (compact). Someone could give me a hint, I do ...
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0answers
41 views

$\mathcal{C}^1$-topology of a submanifold with boundary

Let $M \subset \mathbb{R}^n$ be a compact connected manifold without boundary embedded in $\mathbb{R}^n$, then we can define the $\mathcal{C}^1$-topology of the functions $\mathcal{F}(M)= \{f: M\to \...
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1answer
325 views

Critical Points of a smooth map on a compact manifold

Show that a smooth map $f$ from a compact manifold $N$ to $\mathbb R^m$ has a critical point. (Hint: Let $\pi$: $\mathbb R^m \rightarrow \mathbb R$ be the projection to the first factor. Consider the ...
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0answers
69 views

Calculating Sobolev norm on boundary by extending the map on M as a harmonic map

It is known that the natural trace map $W^{1}(M) \ni \varphi \rightarrow \varphi|_{\partial M} \in W^{1/2}(\partial M)$ is continuous and onto. Since the Dirichlet problem $\Delta \varphi=0$, $\varphi|...
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0answers
46 views

Does this “algebraic” method for the application of the constructive proof of the classification of closed & compact surfaces have any use?

In a typical (at least from what I have seen) geometric topology course, when the classification of closed & compact surfaces is introduced, what has been done first is to find equivalent surfaces ...
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1answer
62 views

Every abelian normal subgroup of a connected and compact Lie group lies in the center

Show that every abelian normal subgroup $H$ of a connected and compact Lie group $G$ lies in the center of $G$. It may be helpfull that if $f:G\to H$ is a surjective lie group homomorphism with $H$ a ...
2
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0answers
42 views

Extending functions on boundary into M as a harmonic function

I am trying to show that if $\varphi \in C^{\infty}(\partial M)$ then there is $\psi \in C^{\infty}(M)$ with $\psi|_{\partial M}=\varphi$ e $\Delta \psi=0$, where $M$ is a compact riemannian manifold ...
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1answer
33 views

Natural surjection that maps loops to cycles

Theorem: If $M$ is a compact Riemannian manifold of dimension $n$ and has nonnegative Ricci curvature, then $b_1(M) \le \operatorname{dim}M = n$, where $b_1$ is the first Betti number. Proof: There ...
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0answers
39 views

Ricci Flow of Surface of Revolution — PDE — Existence and Uniqueness

I consider this article for a numerical approach of the Ricci flow: https://arxiv.org/pdf/math/0406189.pdf%20 and I mainly focus on chapter 3. My three main questions are: 1) Why does boundary ...
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1answer
243 views

Geodesic curvature change under conformal metrics

Suppose that $\sigma_0$ is a fixed metric on a compact riemannian 2-manifold $M$ with boundary $\partial M$. Let $\sigma=\rho \sigma_{0}$, where $\rho=e^{2\varphi}$ with $\varphi \in C^{\infty}(M)$, ...
3
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1answer
43 views

Non-Homeomorphicity of Compact surfaces

Using the fact that every compact surface is homeomorphic to either a connected sum of torii or a connected sum of real projective planes, and using the fact that the corresponding fundamental groups ...
3
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1answer
35 views

Finding a heegaard splitting for general $\sum_g\times I/\phi$

If $\phi$ is an automorphism of $\sum_g$, the closed compact surface of genus $g$, is there a "normal" way to heegaard split $\sum_g\times I/((x,0)\sim(\phi(x),1))$? I was able to find a Heegaard ...
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0answers
34 views

Projection on compact submanifolds

I know from Hilbert projection theorem that for any non-empty, convex and closed set $C$ of a Hilbert space $X$ that there exists a projection map that maps every element in $X$ to a unique element in ...
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1answer
57 views

Why torus space we could see it in $\mathbb R^3$

Anyone know why Torus we can see it in $\mathbb{R}^3$ ? I don't understand why torus as homeomorphic to $S^1 \times S^1$ and see it in $\mathbb{R}^3$, if $ S^1 \times S^1 \subset \mathbb{R}^4$
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1answer
67 views

The number of charts needed to cover a compact manifold

I am learning about differentiable manifold, and I got a question asking to give a reason why I need at least two charts to cover a compact manifold. I know there is a question here also related to ...
2
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2answers
116 views

Compact 3-manifold with trivial first homology

I'm trying to see that a compact 3-manifold $M\subset\mathbb{R}^3$ with $H_1(M)=0$ has $\pi_1(M)=0$. So far: since $M$ is compact, let $\mathcal{T}$ be a smooth triangulation of $M$. I want to be ...
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0answers
169 views

Why the only finite fundamental group of compact surfaces are those of $\mathbb{S}^2$ and $\mathbb{RP}^2$?

I know that the fundamental group of orientable surfaces is of the form $F(a_1,b_1,\ldots,a_n,b_n)/N(\prod a_j*b_j*a_j^{-1}*b_j^{-1})$ where $F$ is the free products with certain generators and $N$ is ...
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0answers
108 views

Orbit space of double torus is triple projective space

By double torus I mean the connected sum of two torus $\mathbb{T}$ denoted $\mathbb{T} \# \mathbb{T}$. By triple projective space I mean the triple connected sum of the projective plane $\mathbb{RP}_2$...
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1answer
137 views

Orbit space isomorphic to Klein bottle

I have proven that the family of homeomorphisms $f_{n,m}(x, y) = (x, (−1)^n y) + (n, m)$ acts properly discontinuously on $\mathbb{R}^2$. Now I should deduce that there exists a covering map of Klein ...
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0answers
81 views

Are there topological manifolds with boundary that are not compact?

Following this question Are there compact manifolds without boundary? I'm asking if there is any example of a manifold with boundary that is not compact, Is the interval $X=[0,1)$ a counterexample ? ...
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1answer
69 views

Why is the boundary of an oriented manifold with its (opposite oriented) copy the empty set?

Excuse the very basic question: I'm following Milnor's Lectures on Characteristic Classes. He defines a relation on the collection of compact, smooth, oriented manifolds of dimension $n$ by letting $...
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2answers
126 views

Why are Hopf manifolds compact?

For a fixed $\lambda>0$, we define a Hopf manifold by the quotient of the group action $\mathbb Z \times \mathbb C \setminus \{0\} \to \mathbb C \setminus \{0\}, (n,z)\mapsto \lambda^n z$. Is there ...
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2answers
101 views

How many ways can we compactify $\mathbb{C}^n$?

Below are two different ways we can compactify $\mathbb{C}$: The first is "adding a point at infinity", the second is "adding a disc at infinity". Intuitively, it looks like these are the only two ...
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3answers
193 views

What is a compact 2-D submanifold?

Show that the qeuations $x^3 + y^3 + z^3 + w^3 = 1$ $x^2 +y^2 + z^2 +w^2 =4$ define a compact 2-dimensional submanifold of $\mathbb{R}^4$. Write the equations for its tangent space at a point $(x_0,...
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1answer
68 views

How to show reflection of $\mathbb{S}^n$ and the identity are not homotopic?

Let $N=\mathbb{S}^n$, $M \subseteq \mathbb{S}^n$ be a hemisphere, including the equator (I am considering $M$ with boundary). Let $f_1:M \to N$ be the inclusion map, and let $f_2:M \to N$ be the ...
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1answer
185 views

Structure of complex Grassmannian $\textrm{Gr}_\mathbb{C}(2,2)$

I recently asked a question about the topology of real Grassmannian $$\textrm{Gr}_\mathbb{R}(2,2) = \frac{O(4)}{O(2)\times O(2)},$$ see Second homotopy group of real Grassmannians $\textrm{Gr}(n,m)$, ...
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1answer
110 views

A question about Gauss-Bonnet theorem on real and complex manifolds

Let $(M,g)$ be a $2$-dimension complex manifold then one can apply Gauss-Bonnet theorem an get some results. On the other hand, this manifold is a real $4$-dimension Riemannian manifold which ...
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1answer
33 views

Smooth structure on $M/G$

Let $M$ be compact smooth manifold and $G$ -- a group acting freely on $M$. The problem is to prove that there is the smooth structure on $M/G$ which makes the map $\pi: M \to M/G$ smooth. I know ...
4
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1answer
111 views

A compact flat manifold whose first Betti number is equal to the dimension is a flat torus

I know the following to be true: If $(M,g)$ is a compact flat Riemannian manifold whose first Betti number ($= \dim H_{dR}^1(M)$) is equal to the dimension, then it is (isometric to) a flat torus. ...