For questions regarding the structure and properties of compact manifolds.

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4
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1answer
47 views

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 $D = \{ (x,x) \in I \times I \}$, ...
1
vote
0answers
14 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 ...
0
votes
1answer
14 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 ...
4
votes
0answers
56 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 ...
4
votes
2answers
194 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 ...
4
votes
2answers
206 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 ...
3
votes
2answers
107 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 ...
1
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1answer
42 views

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 ...
1
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1answer
33 views

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 ...
2
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1answer
54 views

Equivalent definitions of Euler characteristic for closed manifolds

It is well-known that the Euler characteristic of a closed manifold $M^n$, which can be defined as $\chi(M)=\sum_{k=0}^n (-1)^k \operatorname{dim}H^k(M)$, equals the intersection number ...
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0answers
45 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 ...
2
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0answers
54 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
28 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 ...
3
votes
1answer
114 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 ...
1
vote
1answer
55 views

Is a stretched out torus still a $C^\infty$ manifold?

Suppose you have a torus and you carefully make a cylindrical cut down the center. Then you stretch out the outer half and glue together annular regions of the plane in the empty space. Now you have a ...
0
votes
1answer
65 views

Divisors and holomorphic map between a compact Riemann surface and a torus

Let $X$ be a compact Riemann surface (or more generally a compact manifold?) and let $\mathbb{C}^a/\Lambda$ be a complex torus. Suppose we have a holomorphic map $$g: X\to\mathbb{C}^a/\Lambda.$$ By ...
7
votes
0answers
181 views

Showing L infinity norm bounded by L2 norm on a manifold

I have the following problem that I'm working on: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality ...
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0answers
43 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 ...
1
vote
1answer
79 views

Isogenies and dimensions

Let $f: \mathbb{C}^g/L\to\mathbb{C}^{g'}/L'$ be an isogeny of complex tori, i.e. a surjective Lie group morphism with finite kernel. Is it obvious that $g\ge g'$ ? It is easy to show that $f$ is ...
3
votes
1answer
137 views

A literature reference for Sobolev mappings $W^{m,p}(M,N)$ for M, N smooth Riemannian manifolds

Anyone know a respectable reliable reference for the definition of Sobolev mappings $W^{m,p}(M,N)$ for M, N smooth compact Riemannian manifolds. It suffices for m natural and $p\geq 1$
3
votes
4answers
133 views

1-manifold is orientable

I am trying to classify all compact 1-manifolds. I believe I can do it once I can show every 1-manifold is orientable. I have tried to show prove this a bunch of ways, but I can't get anywhere. ...
1
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1answer
59 views

Pullback of support of differential form

Let $f:X \to Y$ be a diffeomorphism between smooth complex compact manifolds. Let $\omega$ be a differential form on $Y$. It is true that the support of $f^*\omega$ is equal to $f^{-1}$ of the support ...
0
votes
1answer
152 views

Degree of composition

Supose that $X\stackrel{f}{\to} Y \stackrel{g}{\to} Z$ are given, $f,g$ smooth and $X,Y,Z$ compact, oriented manifolds. Prove that $$\textrm{deg}(f\circ g) = \textrm{deg}(f)\textrm{deg}(g)$$ where ...
0
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0answers
170 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 ...
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0answers
96 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.
3
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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 ...
1
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0answers
62 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 ...
1
vote
1answer
76 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 ...
4
votes
2answers
86 views

Do there exist cancellable manifolds?

I do not know whether there exists a terminology for that property, but let us say that a closed manifold $C$ is cancellable if for every closed manifolds $M_1$ and $M_2$, $C \times M_1$ and $C \times ...
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1answer
76 views

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

How prove that $\mathbb{CP}^2$ is a compact manifold.
-1
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1answer
63 views

Is O(n) a compact manifold?

I guess O(n) is a compact manifold How can show that it is?
3
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2answers
305 views

A non orientable closed surface cannot be embedded into $\mathbb{R}^3$

Can someone please remind me how this goes? Here's the idea of proof I'm trying to recall: let $S$ be a closed surface (connected, compact, without boundary) embedded in $\mathbb{R}^3$. Then one can ...
11
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1answer
90 views

If the connected sum of a manifold $M$ with itself gives back $M$, does it imply $M$ is a sphere?

Let $M$ be a compact, connected, oriented $n$-dimensional manifold without boundary. Suppose that $M\#M\cong M$. Does it imply that $M \cong S^n$? Sorry if this is a naive question. This is not my ...
1
vote
1answer
67 views

Is the submanifold compact?

Let $M$ be the following subsets of $\mathbb R^4$:$$M= \{(x,y,z,w), 2x^2+2=z^2+w^2, 3x^2+y^2=z^2+w^2 \}$$ we know $M$ is a submanifold of $\mathbb R^4$, is $M$ compact?
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0answers
76 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 ...
4
votes
1answer
48 views

Moving a compact submanifold off of another submanifold?

This is an intuitive idea that I see referenced a lot. Consider the following situation. Let $M$ and $N$ be submanifolds, $M$ compact, in some larger manifold $X$. Suppose also that ...
2
votes
1answer
90 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 ...
1
vote
1answer
95 views

Group Extension and Classifying Space

If $$ 0 \to H \to G \to G/H \to 0\ $$ is a group extension, under what conditions do we have a fibration of the form $$ BH \to BG \to B(G/H), $$ where $BG$ is a classifying space of $G$? Suppose ...
5
votes
4answers
191 views

Is $[0,1]$ a 1-manifold?

Is $[0,1]$ a 1-manifold? I would say no because at either endpoint the open sets containing it aren't homeomorphic to a 1-ball in $\mathbb R^1$.
2
votes
1answer
85 views

Why can an orientation on $X$ be written as a sum of cycles on this open cover?

I'm reading a proof of the following theorem Let $X$ and $Y$ be compact connected oriented $n$-manifolds, and let $f:X\to Y$ be continuous. Let $y\in Y$, and assume that $f^{-1}(y)$ is finite. ...
4
votes
2answers
152 views

Is the additive group of real numbers (R,+) compact?

I have a very naive question about basic topology. My goal is to determine the conditions on a vector field in order for its flow to define a proper (R,+)-action. I understand that if the group G is ...
4
votes
2answers
206 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 ...
3
votes
2answers
82 views

Algebraic surface as a smooth manifold

Let $S$ be the set of points $x=(x_1,x_2,\ldots,x_9)\in \mathbb{R}^9$ which satisfy the following conditions: $$x_1^{2}+x_2^{2}+x_3^{2}=x_4^{2}+x_5^{2}+x_6^{2}=x_7^{2}+x_8^{2}+x_9^{2}=1$$ ...
2
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0answers
66 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 ...
3
votes
1answer
110 views

Approach topological manifolds with smooth manifolds

Because I'm doing some problems that consider all the manifolds while the situation is really clear when considering only smooth manifolds. Thus my question is can we always appoint a topological ...
1
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1answer
43 views

Is $VV^T + D$ a submanifold?

If the positive definite matrix P forms a manifold, is that the subset that {P: P = V V^T + D} where V is a low rank matrix and D is a positive definite matrix a sub-manifold? This idea is ...
7
votes
2answers
265 views

Dimension of de Rham Cohomology groups?

Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
1
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1answer
63 views

set of immersions on a manifold is an open set in the set of all mappings to $\mathbb{R}^{2m+1}$

I'm studying a proof of Takens' Delay Embedding Theorem. A key fact used is that the set of immersions is open in the set of all mappings (mappings from an $m$-dimensional manifold M to ...
3
votes
1answer
119 views

Prove that there are no convex functions on compact manifolds

This one seems intuitively obvious to me but I don't know how to prove it. Suppose you have a compact manifold $M$ with a function $f$ defined on it. Given two points $x$ and $y$ on the manifold, ...
1
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1answer
83 views

Invariant form on Lie algebra

Does anyone have a reference for the following fact? Let $G \subset GL(n)$ be a compact Lie group. Then the form $$f(A)=-Tr(A^2)$$ defined for $A \in T_e G \subset \mathfrak{gl}(n)$ is positive ...