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

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A question on diffeomorphisms and their relation to active coordinate transformations

I've been reading Sean Carroll's notes on General Relativity, http://arxiv.org/pdf/gr-qc/9712019.pdf . I've got to chapter 5 (page 133) and am reading the section on diffeomorphisms in which Sean ...
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41 views

Induced Lie Algebra Representation, Left invariant vector fields and more…

The following is an excerpt from a proof in John Lee's Introduction to Smooth Manifolds I am struggling to understand. I would appreciate if someone was able to help me with whatever it is I am ...
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1answer
38 views

Half-space is not a manifold

We define the half-space $H^n$ as the set containing all tupels $(a_1,\ldots,a_n)$ such that all $a_i\geq 0$. I know that this isn't a manifold - intuitively this is clear - but how can I formally ...
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1answer
40 views

Working with smooth functions defined on a manifold

I am having trouble working with smooth functions defined on a manifold. Is the following line of reasoning valid or not? Let $f$ be a smooth function defined on a manifold $M$ with a local maximum ...
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19 views

Geometric of 3-manifold

I have the 3-manifold $X$ $X=\lbrace(x,y,z)\mid x\neq y \neq z \neq x \rbrace\subseteq S^1\times S^1 \times S^1 $ What is the geometric structure of $X$ from the 8 Thurston Geometries any help ...
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1answer
57 views

Möbius band inside projective plane

How can I see inside the projective plane the Möbius band? I need to know how the Möbius Band appears inside the projective plane. I know it is easy using identifications and algebraic topology. ...
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0answers
43 views

Lie algebras of GL(n,R) and differentials

This question comes from a proof in John Lee's Introduction to Smooth Manifolds, page 194. I am questioning a line in the proof of the following proposition: The composition of the maps ...
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2answers
102 views

What is manifold in Geometry?

What is manifold in geometry? WE always use this word like non-manifold geometry but I was wondering what is manifold in the first place. I got some definition online but couldn't understand. A ...
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28 views

Left invariant vector field question

This question is an exercise from Jeff Lee's Manifolds and Differential Geometry Let $\tilde{X}$ be the left invariant vector field corresponding to $X\in L(V,V)$ (That is, $\tilde{X}_g=(dL_g)_e X$). ...
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28 views

Lie Groups map question

This is a question about Exercise 5.59 in Jeff Lee's Manifolds and Differential Geometry. He writes For a fixed $A\in GL(V)$, the map $L_A:GL(V)\rightarrow GL(V)$ given by $A\mapsto A\circ B$ has ...
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1answer
32 views

Prove that the set of all vector fields $V(S^1)$ is a free $C^{\infty}(S^1)$-module

I need to prove that the set of all vector fields, $X:S^1\to TS^1$ name it: $V(S^1)$, is a free $C^{\infty}(S^1)$-module. So i need a basis $\frac {d}{dx_1},...,\frac {d}{dx_n}$ for $V(S^1)$.It's easy ...
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57 views

Thurston's Geometric structure for 3-manifold

I have an orientable 3-manifold $X$ , such that $$X=\lbrace(x,y,z)\mid x\neq y \neq z \neq x \rbrace\subseteq S^1\times S^1 \times S^1 $$ I would like to know the geometric structure on X. My ...
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32 views

How can we calculate $Xf$?

Let $X$ be the vector field $x \dfrac{∂}{∂x} + y \dfrac{∂}{∂y}$ and $f (x, y, z)$ the function $x^2 + y^2 + z^2$ on $R^3$. Compute $Xf$. Could you give me some hints how we can calculate it?
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1answer
27 views

Outward pointing normal vector to a $k$ manifold in $\mathbb{R}^n $

What do we mean when we say that $\mathbf{N}$ is normal to the the manifold:$$\mathbf{X}:\mathbb{R}^k\to\mathbb{R}^n$$? How do we determine it? How to verify that it's an outward pointing normal?
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3answers
49 views

Length of Circle on a Sphere

I want to calculate the length of the circle $\theta=\pi/4$, where $\theta$ is the latitude, on the unit sphere. I know that the length of a curve $\gamma (t), t \in [0,T]$ on a manifold is given by ...
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0answers
18 views

The orientation of a parametrized $k$-manifold

Can someone explain me what do we mean by "orientation of a $k$-form manifold? Is this definition consistent with the earlier definitions of orientations of curves and surfaces?
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0answers
37 views

Centroid of manifold

The centroid of a compact manifold is defined by the following equation: $c(Y_a)$ is the centroid of the parametrized manifold $Y_a$ is the point in $\Bbb R^n$ whose $i^{th}$ coordinate is given by ...
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34 views

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
17 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
76 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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0answers
37 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
91 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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0answers
19 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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0answers
27 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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0answers
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
47 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 ...
2
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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
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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 ...
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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
367 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
44 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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64 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
51 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
22 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
20 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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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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0answers
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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
62 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
36 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
31 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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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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31 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
26 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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0answers
63 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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0answers
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
221 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
49 views

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 ...