For questions regarding the structure and properties of compact manifolds.

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23 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
19 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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1answer
30 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
43 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 ...
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0answers
13 views

Integrating over a flag manifold

I need to calculate an integral over the flag manifold $U(4)/U(1)\times U(1)\times U(1)\times U(1)=U(4)/T^4$. How can I derive the correct Haar measure to use?
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1answer
59 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 ...
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101 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
30 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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1answer
78 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 ...
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1answer
70 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 $p\geq 1$
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4answers
93 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. ...
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1answer
43 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 ...
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1answer
80 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 ...
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0answers
77 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
50 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
83 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
42 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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1answer
65 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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2answers
78 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
56 views

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

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

Is O(n) a compact manifold?

I guess O(n) is a compact manifold How can show that it is?
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2answers
182 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 ...
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1answer
86 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 ...
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1answer
64 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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46 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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1answer
42 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 ...
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1answer
75 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
80 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 ...
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4answers
162 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 $\mathbb R^1$.
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1answer
78 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. ...
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2answers
136 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 ...
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61 views

Show that $\int_Md\omega=0$.

Let $\omega$ be a continiously differentiable $(k-1)$-form in the open set $U\subset\mathbb{R}^n$ and $M\subseteq U$ an orientated compact k-dimensional manifold. Show that $$ ...
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1answer
127 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 ...
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2answers
74 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: $$x_1^{2}+x_2^{2}+x_3^{2}=x_4^{2}+x_5^{2}+x_6^{2}=x_7^{2}+x_8^{2}+x_9^{2}=1$$ ...
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0answers
53 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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1answer
96 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 ...
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1answer
41 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 ...
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2answers
183 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?
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1answer
47 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 ...
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1answer
98 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, ...
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1answer
76 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 ...
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1answer
108 views

a theorem in topology

Is there anyone know there is a theorem in topology which states that a compact manifold "parallelizable" with N smooth independent vector fields. must be an N-torus? and why the vector field here is ...
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1answer
205 views

Schrödinger Kernels on manifolds

Let $M$ be a compact Riemannian manifold and $\Delta$ be the Laplace-Beltrami operator. It is well-known that the solution operator to the heat equation $e^{t \Delta}$ is smoothing for $t>0$ and ...
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1answer
137 views

Every compact orientable surface with $S^1\times\{0\}$ as its boundary intersects the $z$-axis

Let $M$ be a compact orientable surface (manifold in $\mathbb R^3$) with boundary $S^1\times\{0\}$.Show that $M$ intersects the $z$-axis. Some ideas: $1)$Since $M$ is a compact orientable manifold ...
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1answer
286 views

Uniformization Theorem for compact surface

Why in proof of proposition 6 of http://arxiv.org/abs/0909.1665, they claim that if a embedded surfaces $\Sigma^2 \subset (M^3,g)$ is homeomorphic to $\mathbb{RP}^2$, where $M$ is compact manifold, ...
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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 ...