# Tagged Questions

For questions regarding the structure and properties of compact manifolds.

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### Bounding a Sobolev norm on product manifold

Suppose you have a closed manifold $M$ and a function $f:M\times M\rightarrow\mathbb{C}$ that is $C^2$ in the $M$ variables, and suppose that at any point $q\in M$, we have $$f(\cdot,q)\in H_m(M).$$ ...
2answers
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### Suppose $M$ has trivial 1-st de Rham cohomology group. For which integers $k$ does there exist a smooth map $f : M → T^n$ of degree $k$?

Let $M$ be a compact oriented smooth $n$-manifold, with $H_{dR}^1(M)=0$. For which integers $k$ does there exist a smooth map $f : M → T^n$ of degree $k$? I know that if $M$ is simply-connected, we ...
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### integral constraint induce a manifold on Sobolev space

given the set $$M:=\{u\in H^2(\Omega):\int_{\Omega}u=m\,\}$$ $m\in \mathbb{R}$, $\Omega$ is a bounded piecewise smooth domain in $\mathbb{R}^n$. also denote by $u(t)$ a map: $u(t):(0,T)\to M$ ...
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### Finding the boundary of the continuous image of a compact simply-connected Lie group

Statement of the problem Given a continuous map $f:G \rightarrow D^2$ where $G$ is a compact simply connected Lie group and $D^2$ is the unit disk in the plane, I have shown that: There exists a ...
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### Hypersurfaces of a hemisphere

Let $S_+^{n+1}$ be the open hemisphere of the standard euclidean sphere centered at the north pole and let $M^n$ be a compact, connected and oriented hypersurface of $S_+^{n+1}$. Is it true that if $M$...
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### Surjectivity of the exponential map on SO(2n)/U(n)

Let $M:=SO(2n)/U(n)$ the homogeneous space of all orthogonal almost-complex structures on $\mathbb{R}^{2n}$. When $n=2$, it is known that $M$ is just the 2-sphere. 1) On the 2-sphere, the ...
1answer
49 views

### Finitely Many genus-g Quotients of Compact Riemann Surface

I hear there is a semi-famous theorem from my advisor, but he didn't know the name and I was unable to find it online. Does anybody know of the following? Let $S$ be a compact Riemann surface. Then ...
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### connected sum of surfaces is well defined proof attempt

Suppose $S_1$ and $S_2$ are compact surfaces (connected 2-dimensional manifolds). If we cut out of them two closed disks, and glue the surfaces along disk boundaries we get new surface, their ...
0answers
31 views

### Why are PDEs with Hamiltonians usually solved on compact manifolds?

The title is self explaining: I see in a lot of literature that PDEs with some Hamiltonian structure in it are solved over a torus or some other compact manifold. Why is that? At least I now that it ...
1answer
63 views

### Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$?

The first question is Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$? In other words, is it obvious? The question stems from a theoretical physics ...
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### What is the role of the Laplace-Beltrami operator in providing an optimal embedding for a manifold?

I ve read in "Laplacian eigenmaps for dimensionality reduction and data representation" of Belkin and Niyogi that the Laplace-Beltrami operator has a role in providing an optimal embedding for the ...
1answer
27 views

### How to qualify a N dimensional manifold as Compact under following condions?

Suppose a manifold of N dimensions is closed and bounded in a dimension but it remained unbounded in all other dimensions, so how to categorize the manifold. For example, in simpler form how to ...
0answers
43 views

### Fundamental group of result of 0-Dehn surgery and meridian

Let $M_K$ be the result of $0$-Dehn surgery along a knot $K$ in $S^3$. Let $m$ be a meridian to $K$, and we view $m$ as a circle in $M_K$ (without changing the notation for it). The claim I wanted to ...
3answers
411 views

### Examples of manifolds that are not boundaries

What are some examples of manifolds that do not have boundaries and are not boundaries of higher dimensional manifolds? Is any $n$-dimensional closed manifold a boundary of some $(n+1)$-dimensional ...
1answer
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### Showing that $\mathbb S^1$ is a deformation retract of the Mobius strip, rigorously.

Intuitively, I can see why this is. I've found a few threads about this, but they only provide, for example, a deformation retraction of $I \times I$ to its diagonal $D = \{ (x,x) \in I \times I \}$, ...
0answers
27 views

### Reference needed for a short time existence result of quasilinear PDE on a compact manifold (relating to Ricci flow).

I'm currently in the proces of learning and writing a bit about the Ricci flow. In particular I'm studying the case of compact 2d Riemannian manifolds. Mostly I'm making good progress but I do miss ...
0answers
69 views

### Map of smooth manifolds

Let $M$ and $N$ be smooth, connected $n$-dimensional manifolds. Let $M$ be compact and non-empty. Show that every embedding $f: M \to N$ is a diffeomorphism. So because $f$ is a embedding we have ...
1answer
111 views

### Is the orbit map for a group action closed in this case?

Suppose a compact Lie group $G$ acts on a manifold $M$ and let $\pi : M \rightarrow M/G$ be the orbit map. Can I say that $\pi$ is closed map? If $C \subseteq M$ is a closed set in $M$ then I only ...
2answers
234 views

### Integrals of Pullbacks

This is a problem from Guillemin's Differential Topology: Suppose that $f_0, f_1: X \to Y$ are homotopic maps and that the compact boundaryless manifold $X$ has dimension $k$. Prove that for all ...
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237 views

### How to Show Cotangent Bundles Are Not Compact Manifolds?

Hamiltonian mechanics occurs in a sympletic manifold called phase space. Lagrangian mechanics take place in the tangent bundle of the configuration manifold. Using Legendre transform makes possible ...
2answers
266 views

### Why spheres are not symplectic manifolds?

Reading some books on diferential geometry, a found that $S^{2n}$ (with $n>1$) are not symplectic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not ...
1answer
79 views

### How prove that $\mathbb{CP}^2$ is compact? [closed]

How prove that $\mathbb{CP}^2$ is a compact manifold.
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### Can a product of a Stein manifold and a compact manifold be again Stein?

A Stein manifold is a manifold which is holomorphically separable and convex. It is well known that a product of two holomorphically convex (resp. Stein) manifolds is again holomorphically convex (...
1answer
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### Topology of Manifolds and Kunneth fomula by Griffiths Harris

Reading Griffiths-Harris, at page 56 I find some parts that I can't understand. 1) After having proved the Poincaré Duality Theorem, Griffiths and Harris proceed to prove a weaker result, that's to ...
2answers
303 views

### What is group manifold of a compact Lie Group?

I searched on google the meaning of a group manifold of a compact lie group but I didn't get the answer. A paper on arxiv "Background Independent Quantum Gravity:A Status Report- Abhay Ashtekar" on ...