Questions tagged [compact-manifolds]
For questions regarding the structure and properties of compact manifolds.
1
vote
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 ...
3
votes
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 $\...
4
votes
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 ...
1
vote
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 $(\theta,\varphi)$, ...
0
votes
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?
0
votes
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' ...
0
votes
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. ...
1
vote
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 ...
1
vote
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 ...
0
votes
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{...
0
votes
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, ...
2
votes
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
votes
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 ...
2
votes
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 ...
2
votes
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 $\...
-1
votes
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?
3
votes
1answer
34 views
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 ...
1
vote
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
votes
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 \...
0
votes
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 $...
8
votes
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)...
1
vote
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 ...
1
vote
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 \...
5
votes
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 ...
3
votes
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|...
1
vote
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 ...
0
votes
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
votes
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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
votes
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
votes
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 ...
1
vote
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 ...
0
votes
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$
1
vote
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
votes
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 ...
0
votes
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 ...
5
votes
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 $\mathbb{RP}_2$...
1
vote
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 ...
0
votes
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 ? ...
2
votes
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 $...
0
votes
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 ...
0
votes
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 ...
2
votes
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,...
0
votes
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 ...
3
votes
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 $\textrm{Gr}(n,m)$, ...
0
votes
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 ...
0
votes
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
votes
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.
...
