For questions regarding the structure and properties of compact manifolds.

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2answers
56 views

How can I show $R^n$ is dense in $S^n$?

How can I show $R^n$ is dense in $S^n$? I wanted to show $S^n$ is compactification of $R^n$. for this I need $R^n$ is not compact, for this there is no problem, and $S^n$ is compact, I did it with ...
3
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1answer
67 views

Momentum is quantised in compact spaces?

Background One of the first examples given when studying quantum mechanics is the particle on a cylinder, or particle on a ring. One finds that because of the periodic boundary conditions, ...
3
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1answer
227 views

Proof that a map from an orientable surface to a non-orientable surface has even degree.

For a smooth map $f:M\to N$ from an orientable closed surface $M$ to a non-orientable closed surface $N$, we define its parity (also called modulo 2 degree, and denoted $\deg_2(f)$) as the parity of ...
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1answer
36 views

Coverings of a three-manifold

He guys, I have two questions regarding the following: Consider the three-manifold $\mathbf{T}^3 = S^1 \times S^1 \times S^1$ and let $S_n$ be the permutation group acting on $n$ letters. Let $\phi:\...
2
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1answer
105 views

orientation of symplectic manifold and lagrangian submanifolds

A statement: The self-intersection index of lagrangian submanifold $M \subset X$ is equal to Euler characteristic $\chi(M)$. How I should oriented $X$? Let's consider some example. The null-section $...
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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 ...
1
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1answer
81 views

Partition of unity. Does this one exist?

Let $X:=\mathbb{R^n}$ be given and $M \subset X$ be a compact set in it. Then my question is: Are there $\alpha_i \in C^{\infty}(X,\mathbb{R})$ such that $supp(\alpha_i) \subset N$, where $N$ is an ...
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1answer
45 views

When are flag manifolds compact?

This is a question from Lee's book on Smooth Manifolds, question 21-16: Let $F_K(V)$ be the set of flags of type $K$ in a finite-dimensional (real) vector space $V$. Show that $GL(V)$ acts ...
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0answers
314 views

Norm Inequality on a Compact Riemannian Manifold

Consider the following problem: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality $$\Delta u \geq -...
4
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0answers
68 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 ...
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0answers
84 views

Measurability of points regular

I'm reviewing the proof of the theorem of oseledet the book Mañe: Let $M$ a compact metric space and $f:M \rightarrow M$ a homeomorphism, $\pi: F \rightarrow M$ a finite-dimensional continuos vector ...
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0answers
38 views

Total Gaussian curvature

For a compact surface, $S$, in $\mathbb{R}^3$, how would I go about showing that the total Gaussian curvature $\int_S K da \leq 4 \pi$? I feel like Hopf's Umlaufsatz and the Gauss-Bonnet Theorem are ...
2
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0answers
38 views

A very general question about blow-up for experienced symplectic topologists and algebraic geometers

I am trying to understand the process of symplectic blow-up of compact symplectic manifolds $M^{2n}$ along compact symplectic submanifolds $X$ on a deeper level, and I am also searching for relevant ...
2
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0answers
91 views

Is log-general type an intrinsic property of a variety

Let $X$ be a smooth quasi-projective variety over $\mathbb C$. Let us say that $X$ is of log-general type if for some choice of smooth compactification $\bar X$ with normal crossings boundary divisor ...
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0answers
80 views

Standards in P.L. Topology

About a week ago, the reading course on PL topology I'm going to follow started. The aim of the reading course is to understand the basics of PL topology and have a reasonable to good understanding of ...
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0answers
19 views

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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0answers
51 views

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 ...
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0answers
54 views

A lemma in Milnor's book “Topology from the Differentiable Viewpoint”

In preparation for proving the Poincaré-Hopf Theorem, Milnor states in his book (see p. 38, Theorem 1) For any vector field $v$ on $M$ with only nondegenerate zeros, the index sum $\sum \iota$ ...
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0answers
30 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 ...
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0answers
59 views

Compact 1-manifolds

I want to find all compact 1-manifolds. I think these will be lines, with fixed endpoints, and also the case where the endpoints meet. Hence I think all compact 1-manifolds are homeomorphic to the ...
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0answers
42 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 ...
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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 ...
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0answers
81 views

Lens Space Orientation Reversing Homeomorphism

I am thinking of an example where the connected sum of two three Manifolds depends on the chosen orientation. Hempel gives in his book "3-Manifolds" an example, namely lens spaces. He shows that two ...
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0answers
29 views

Embedding of manifolds of constant negative curvature

Consider the manifold of constant negative curvature $G=SL(2, R)/\Gamma$ where $\Gamma$ is such that $G$ is compact (I have no special constraint on $\Gamma$). I know that by the Whitney embedding ...
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0answers
87 views

Why is the Compact Symplectic Group Simply Connected

Let $Sp(n)=U(n,\mathbb{H})=\{A \in M_{n}(\mathbb{H}) : A\cdot A^{*}=I\}$ be the compact symplectic group, a subset of $Sp(2n,\mathbb{C})$. I want to show that $Sp(n)$ is simply connected, in ...
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0answers
122 views

Poisson equation on a Torus

I need an example for a Poisson equation ($\triangle_S u = F $) on a torus. Specifically, i will appreciate a function F and its corresponding analytic solution ($u_{ex}$). Any reference will be ok.
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0answers
74 views

Compactness and Poincare duality

I am reading Appendix B in Fulton's Young Tableaux about Borel-Moore homology. In particular, I'd like to understand why for compact manifolds the Borel-Moore homology groups are isomorphic to ...
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0answers
114 views

Triangulation of a 3-sphere

If one wants to generate a Simplicial complex of the topology of the 3-sphere, one can just take the boundary of a 5-cell, 16-cell or 600-cell. The curvature is concentrated on the edges meeting the ...
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0answers
81 views

Geodesic On Compact Manifolds

Let $M$ be a compact Riemmanian manifold. Let $G$ denote the set of all geodesics of $M$. If $\gamma\in G$ let $l(\gamma)$ denote its length. Let $$S=\sup\{l(\gamma): \gamma\in G\}$$ Suppose $S<\...
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0answers
46 views

Is there a preferred way to characterize a cone in cohomology?

In a Euclidean space, one can of course describe a cone using a generalization of polar coordinates, and since the de Rham cohomology spaces are themselves Euclidean spaces, one can do the same here. ...
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0answers
12 views

About the number of minimum parametrizations of a $1$-smooth manifold compact w/ boundary in $\mathbb{R}^{3}$

Let $C$ be a $1$-dimensional compact differentiable manifold with boundary in $\mathbb{R}^{3}$. In easy examples, it looks like we can always parametrize such a manifold with only two charts: usually ...
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0answers
24 views

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 ...
0
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0answers
291 views

Euler characteristic of the product

I want to prove that Euler characteristic of the product of two compact oriented manifolds is the product of their Euler characteristics. As always I do, I'm considering Guillemin-Pollack definitions,...