For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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Relative singular homology $H(M,\partial M)$ for a manifold $M$?

Let $M$ be an orientable manifold. What can be said about the relative homology $H(M,\partial M)$? Perhaps one can calculate the homology using excision?
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1answer
14 views

Geometric interpretation of the evaluation of Poincaré dual with a fundamental class

Given oriented, closed submanifolds $X^k$ and $Y^{n-k}$ in an oriented, closed $n$-manifold, is there a nice geometric interpretation of the evaluation $\langle \operatorname{PD}([X]),[Y]\rangle$? I ...
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1answer
59 views

connected manifolds are path connected

prove every connected manifold is path connected manifold . my thought: connected space : Let $ X$ be a topological space. A separation of $ X $ is a pair $U, V$ of disjoint nonempty ...
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35 views

How to find the tangent space of $O(n)$ by considering $O(n)$ as the pre-image of the map $A \mapsto AA^T$ at identity?

Why is the tangent space of $O(n)$ at $H$ equal to $T_H O(n) = \{ M \in \mathbb{M} ( n, \mathbb{R} ): (DF(H))(M) = 0 \}$, where $$F: \mathbb{M} ( n, \mathbb{R} ) \cong \mathbb{R}^{n^2} \to ...
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2answers
89 views

Interior of a compact 3-manifold

I have an orientable 3-manifold $X$, such that $$X=\lbrace(x,y,z)\mid x\neq y \neq z \neq x \rbrace\subseteq S^1\times S^1 \times S^1 $$ How to find a compact 3-manifold $M$ such that $X= ...
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17 views

Does the integral over a covariant derivative give zero on curved manifolds?

On a flat manifold like $M=(g,\mathbb{R}^n)$ with Euclidean metric $g$ one has the simple result that an integral over the derivative of a vector field $X^a$ gives zero, if $X^a$ behaves well at the ...
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25 views

Are Random variables in delay embedded phase space independent?

Consider a smooth manifold $M=R^d$ embedded in a higher dimensional space $R^D$ using Takens Attractor reconstruction. Let, the Random Variable $X \in R^d$ have a Gaussian pdf and the random variable ...
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21 views

What geometric information can be recovered from $L^2(X)$ for a manifold $X$?

It is well known that a compact Hausdorff topological space can be fully reconstructed from its $C^\ast$-algebra of complex valued continuous functions with the sup norm. Are there similar (partial) ...
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1answer
46 views

Changing $[0,2\pi)$ with $S^1$ such that a map defined on $[0,2\pi)$ stays unchanged

* Consider the following procedure of changing the domain of a map, but the map remaining essentially the same - illustrated, for concreteness, in case of the polar-coordinates map.* Let ...
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1answer
31 views

Covariant Stared Vector Spaces?

I reading john Lee's book entitled "Introduction to Smooth Manifolds" and on page 311 $$T^{k}\!\!\left(V^{*}\right)=V^{*}\!\!\times\!\!V^{*}...V^{*}\;\;k\text{-times}$$ is defined as the vector ...
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1answer
54 views

$G$ is an $(n-1)$-manifold without boundary and is the topological boundary to an open $K\subset \mathbb{R}^n$. Prove $G \cup K$ is an $n$-manifold.

All manifolds are smooth. Let $M = G \cup K$. The interior of $M$ is an open set in $\mathbb{R}^n$ and can be given a global coordinate by the identity map. The points in $M$ not on the interior of ...
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2answers
71 views

What's the “easiest” closed 3-manifold with a nonabelian fundamental group?

I'm looking for some easy compact, oriented 3-manifolds without boundary that have a nonabelian fundamental group. It needn't be perfect. "Easy" means that it has an easy Heegard diagram, say, one ...
2
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1answer
37 views

Lie Subgroup Example - Explanation?

I'm currently working through Jeff Lee's 'Manifolds and Differential Geometry'. He defines a Lie Subgroup, $H$, to be an abstract subgroup of a Lie Group $G$, such that the inclusion map ...
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2answers
324 views

Proof of whitney's embedding theorem?

While learning about the rigorous definition of manifolds, my text mentions that any $n$-dimensional manifold can be embedded in $\Bbb{R}^{2n}$, which is called Whitney's embedding theorem. I have ...
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1answer
43 views

Polynomials of a fixed degree have a nonzero partial at all points of their vanishing set?

I'm having difficulty with the second half of a question from an old homework assignment (for a differential geometry class I am currently taking). The first half of the question asked me to assume ...
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2answers
66 views

Prove $\omega\wedge\eta=0$ iff there exist a $\theta\in A^3(\mathbb{R}^5)$ such that $\eta=\omega\wedge\theta?$ [closed]

Suppose $\omega\in A^1(\mathbb{R}^5)$ and $\eta\in A^4(\mathbb{R}^5)$. prove $\omega\wedge\eta=0$ iff there exist a $\theta\in A^3(\mathbb{R}^5)$ such that $\eta=\omega\wedge\theta?$
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62 views

Generalizations of the Hairy Ball Theorem to wider classes of manifolds

In 2 dimensions the hairy ball theorem generalizes from spheres to all orientable closed manifolds with nonzero Euler characteristic. The hairy ball theorem holds for all even dimensional ...
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1answer
40 views

Using Alexander's Theorem to show that the sphere $S^3$ is a prime manifold

I'm completely aware of the triviality of this question, but for some reason, I can't visualize the argument. In Hatcher's 3-manifold notes, the form of Alexander's theorem stating that Every ...
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0answers
18 views

Partition of unity and coordinate patches

I have a question related to terminology. Assume that $M$ is a $k$ manifold. What does it mean to say that the partition of unity $f_1,f_2,...f_n$ on $M$ is dominated by the collection of all ...
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18 views

Embedding of classical Lie groups

This is somehow very natural question so I'm sure that the answer should be well known: Whitney theorem states that each (say paracompact) $n$-dimensional manifold could be embedded in ...
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0answers
17 views

Smooth sections of smooth vector bundle

Suppose that $E \to B$ is a (real for example) smooth vector bundle ($B$ is assumed to be a smooth manifold). There is a important notion of the smooth section $s:B \to E$: is has to satisfy $s(x) \in ...
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31 views

A transversal argument to show the dependence of the intersection number of two cycles from the homology class

At page 51, Griffiths and Harris are going to prove the Poincaré duality theorem, so they have to define the intersection number of two cycles $A$ and $B$ on a manifold. In order to show the ...
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20 views

Question about Nehari manifold

I have two questions about the Nehari manifold, the first one is where i can found the properties of this manifold ? and the second one is in a theorem it says "let M a Banach-finsler manifold" can ...
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1answer
57 views

Evaluating contractions of a tensor product

Consider $T = \delta \otimes \gamma$ where $\delta$ is the $(1,1)$ Kronecker delta tensor and $\gamma \in T_p^*(M)$, the co-tangent space over some manifold $M$. Evaluate all possible contractions of ...
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1answer
34 views

Vector Fields on Real Numbers

I'm looking at vector fields on the manifold $\mathbb{R}$, in the sense that a vector field $v$ is a smooth function from $\mathbb{R}$ to $\mathbb{R}\times T_p\mathbb{R}$. These seem so simple that ...
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1answer
30 views

Differential: Smoothness

This is a lemma for another thread. Given smooth manifolds. The differential is a smooth map: $$F:M\to N:\quad F\text{ smooth}\implies\mathrm{d}F\text{ smooth}$$ How to check this?
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67 views

Question about Milnor's proof of Sard's Theorem

We've just covered Sard's theorem and have just started to look at transversality in my differential geometry class and I'm trying to understand a proof of Sard's theorem (based on Milnor's proof): If ...
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0answers
28 views

Open cover of manifold with boundary

If $\mathcal{O}=(U_{\alpha})_{\alpha\in A}$ is an open cover for a smooth manifold $M$, then each $U_{\alpha}$ is a smooth manifold. I want to extend this fact to manifolds without boundary. So my ...
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2answers
25 views

What kind of manifold is a configuration manifold?

I have recently been learning about the basic properties of topological, smooth, and Riemannian manifolds. But I frequently hear the term configuration manifold referenced in relation to Lagrangian ...
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60 views

When did the notion of the tangent space emerge?

When did the modern conception of/notation for the tangent space to a manifold come into use?
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19 views

Charts/transition charts of the $\mathbb{CP}^3$ tangent bundle

I would like to explicitly compute the charts and transition charts for the tangent bundle of $\mathbb{CP}^3$. I know the charts of $\mathbb{CP}^3$ are $\phi_i: U_i=\{[z_0,z_1,z_2,z_3]; z_i \neq ...
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2answers
218 views

Equivalence of two distance function on a Riemannian manifold

Let $(M,g)$ be a closed connected $m$ dimensional smooth Riemannian manifold and assume that it is isometrically embedded in a Euclidean space $\mathbb{R}^q$ by $\iota:M\to\mathbb{R}^q$. $|\ast|$ ...
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0answers
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Smooth manifolds having an analytic structure.

I am reading about Manifolds and Lie groups and I have certain questions related to them. Let me first explain why I am asking these questions. We know that not all $C^{\infty}$ functions are ...
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Decorating eggs

I came across this question as I have been helping someone decorate styrofoam eggs by gluing wrapping paper onto it. The question is quite simple but I found myself stumped. Given an egg what is the ...
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2answers
198 views

Why is Klein bottle non-orientable?

I am doing the homework of differential geometry and encounter this problem: The Klein bottle $K^2$ is defined to be the identification space $$[0, 1] \times [0, 1]/{\sim}, \text{ where the ...
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38 views

Does isometry preserve volume on open sets?

Suppose there are two open sets $A,B$. $h$ is an isometry. And the function $h$ maps $A$ to $B$; $h(A)=B$. I need to show that isometry is volume preserving. Any hint would be appreciated! Thanks ...
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2answers
62 views

The quotient map and isomorphism of cohomology groups

Let $X$ be a closed $n$-manifold, $B$ an open $n$-disc in $M$. Suppose $p:X\rightarrow X/(X-B)$ is a quotient map. Notice that $X/(X-B)$ is homeomorphic to the sphere $\mathbb{S}^n$. My question is ...
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0answers
30 views

Tangent space of a Product of two manifolds

Suppose $M$ and $N$ are two $C^\infty$ manifolds. Take $p\in M$ and $q\in N$. We have the following maps between these: $\iota_1 : M\to M\times N$, $\iota_2:N\to M\times N$, $\pi_1:M\times N\to M$ and ...
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0answers
31 views

To Prove that The Level Set Of AConstant Rank Map is a Manifold

Let $f:\mathbf R^n\to\mathbf R^m$ be a smooth function of constant rank $r$. Let $\mathbf a\in \mathbf R^n$ be such that $f(\mathbf a)=\mathbf 0$. Then $f^{-1}(\mathbf 0)$ is a manifold of ...
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1answer
91 views

Why do we require that a complex manifold has the structure of a real manifold?

I am taking a course in complex manifolds, heavily influenced by Huybrechts' book "Complex Geometry", and in it we define a complex manifold $X$ to be a smooth, real manifold $M$ together with an ...
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35 views

What does matrix decomposition really mean?

Any element of the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ can be decomposed using the Euler decomposition into the product of three matrices. \begin{equation} S = O\begin{pmatrix}D & ...
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2answers
139 views

Invertibility theorem on the boundary for a function between two closed 2D manifolds

Assume a function $f:\mathbb{R}^2\to\mathbb{R}^2$ on a simply connected, closed domain $D\subset\mathbb{R}^2$ including its boundary $\partial D$. I am interested in the local invertibility of $f$ ...
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57 views

Is there a way to define the concept of manifolds so it looks more like “generalised affine spaces”?

What I have in mind is along the lines of this: Let $M$ a topological space, $V$ a normed vector space, and $$ \boxminus \colon M\times M \to V, $$ $$ \boxplus \colon M\times V \to M. $$ Then ...
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1answer
77 views

Complex projective manifolds and smooth projective varieties

Look at the following theorem: The following two categories are equivalent: The category of non-singular projective varieties over $\mathbb C$. (Where a variety is understood as in ...
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1answer
33 views

How to prove that volume forms agree on $U_\alpha \cap U_\beta$?

I am familiarizing myself with Riemannian manifolds. Let $M$ be an orientable smooth $n$-manifold with atlas $(U_\alpha, (x_1^\alpha, \dots, x_n^\alpha))$ and let $g$ be a Riemannian metric on $M$. ...
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2answers
65 views

Stokes or homotopy?

The problem states Show that if $X$ is a simply connected manifold, then $\oint_{\gamma}\omega=0$ for all closed 1-forms $\omega$ on X and all closed curves $\gamma$ in $X$. However I have ...
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2answers
51 views

Question concerning tensors

As some of you may have seen from my previous question(s), I am working through Spivak's Calc on Manifolds and this happens to be the first time I've been introduced to tensors formally, though I have ...
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1answer
69 views

Gauß and mean curvature

I was wondering whether the Gauß, mean curvature and shape operator of a surface actually depend on the chosen parametrization? Under a reparametrization of $f: \Omega \subset \mathbb{R}^2 ...
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1answer
118 views

Isometric and conformal map

We defined conformal and isometric maps for surfaces $f,g: \Omega \subset \mathbb{R}^2 \rightarrow S \subset \mathbb{R}^3$. Under a reparametrization of $f$ I understand a diffeomorphism $\Phi : M ...
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Function represented as composition

Question:Prove that if $\vec{g} : \mathbb{R}^n \rightarrow \mathbb{R}^n $ and $ \det(\vec{g}') \neq 0$, then in some open set $V \subset \mathbb{R}^n $ such that $\vec{x} \in V$ we have: $\vec{g} = ...