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

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1answer
39 views

Dependence on the class of differentiability between manifolds and maps

Maybe a silly question, but in some books (like "Differential Geometry - Manifolds, Curves, Surfaces - Gostiaux and Berger"), when differentiable maps of class $C^s$ are defined, we have something ...
3
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1answer
103 views

Integral curves of $X = z \dfrac{\partial}{\partial \theta} - \sin \theta \dfrac{\partial}{\partial z}$ on a cylinder

Consider coordinates $(\theta, z)$ on $S^1 \times \mathbb R$, and a vector field $$X = z \dfrac{\partial}{\partial \theta} - \sin \theta \dfrac{\partial}{\partial z}.$$ Show that the integral curve of ...
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2answers
160 views

de Rham cohomology of $\mathbb R^2 \setminus \mathbb Z^2 $

I am trying to calculate the cohomology of $X = \mathbb R^2 \setminus \lbrace \mathbb Z \times \mathbb Z \rbrace = \lbrace (x,y) \in \mathbb R^2 : x \text{ and } y \not \in \mathbb Z \rbrace.$ ...
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0answers
50 views

Definition of Manifold's Orientation

I am reading the book of manifold. And I find there are many definitions about one object, such as orientation, Euler character and degree of map. I am confused with the conception of orientation. ...
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0answers
58 views

Tanget space to manifold via curves without map

I define tangent space T to differentiable manifold, in point p, via equivalence class of curves. The condition for this equivalence is $(\varphi\circ\gamma_1)'(0)=(\varphi\circ\gamma_2)'(0)$ for some ...
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0answers
57 views

$\mathbb{RP}^2$ does not embed into $\mathbb{R}^3$: reduction to the differentiable case

It is not difficult to see that the real projective plane cannot be embedded into $\mathbb{R}^3$ as a differentiable submanifold (for example one can easily show that the complement would consist of ...
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3answers
96 views

A subset $A$ of a manifold $X$ that is a manifold but not a submanifold of $X$

Let $X$ be a manifold and $A$ a subset of $X$. Is it possible for $A$ to be a manifold without being a submanifold of $X$. Thank you for your help!!
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0answers
30 views

Proving a global property of the connection from a property of local connection matrices

Let $D$ be an $m$-dimensional distribution on an $n$-dimensional manifold $M$. Let $U\subset M$ be an arbitrary open subset such that on $U$ we can define vector fields $e_i$ such that for each $x\in ...
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0answers
47 views

Tensor vector bundle construction

$\newcommand{\p}{\partial}$Let $M$ be a smooth manifold, and define $$T_{r,s} := \bigsqcup_{p \in M} (T_p M)_{r,s} = \bigsqcup_{p \in M} \big( \underbrace{T_p M \otimes \dots \otimes T_p M}_r \otimes ...
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0answers
98 views

Every element of the tangent space of a manifold $M$ is the tangent to a smooth curve in $M$.

I am reading the first chapter on Manifolds from the book Warner, in which I have the following doubt. Let $M$ be a differentiable manifold. A $C^{\infty}$ mapping $\sigma:(a,b)\longrightarrow M$ is ...
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1answer
31 views

Involutive Properties of Space-structures on Smooth Manifolds

I am currently reading Quantum Invariants of Knots and 3-Manifolds by Turaev, and I am having a hard time understanding a statement made on page 120. He is explaining the property of space-structures, ...
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1answer
131 views

Pullback of a form using the Hopf fibration

I embed the 2-sphere and the 3-sphere in $\mathbb{R}^3$ and $\mathbb{R}^4$ respectively. Then denote by $\{x_1,x_2,x_3,x_4\}$ the coordinates on the 3-sphere and $\{y_1,y_2,y_3\}$ on the 2-sphere. So ...
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1answer
119 views

Geodesics of one-dimensional manifold

I apologize if my post is "silly" because I don't know much about riemannian geometry. I know that $M = (0,1)$ (the open unit interval) can be seen as a one-dimensional manifold. Since $M$ is an ...
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1answer
40 views

Lie brackets definition

Let $v,w$ be vector fields on a smooth manifold $M$ (i.e. $v : M \rightarrow TM = \lbrace (p, v_p) : p \in M, v_p \in T_p M \rbrace$). The Lie brackets of $v,w$ are defined as $$ [v,w](f)|_p = ...
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0answers
68 views

Structure equations on the 3-sphere

On the 3-sphere I have found the vector fields $X_1=(-x_2,x_1,-x_4,x_3)$, $X_2=(-x_3,x_4,x_1,-x_2)$, $X_3=(-x_4,-x_3,x_2,x_1)$, in the basis $\left\{\frac{\partial}{\partial ...
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0answers
150 views

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
4
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0answers
96 views

Flowing a vector along a vector field $X$ using the pushforward of the flow of $X$

On the three-sphere $S^3$, I'm given three vector fields $X$, $Y$ and $Z$, such that at each point $p\in S^3$, the tangent vectors $X_p$, $Y_p$ and $Z_p$ form an orthogonal basis of the tangent space ...
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3answers
84 views

Embeddings (how to prove them exactly)

For which of the following sets is the statement: '$A$ can be embedded in $B$' true? I can try to decide this intuitively but don't know if I'm right, and surely don't know how to formally prove it. ...
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2answers
86 views

Surface of graph of a function.

Let $G:=\{(x,y,z)\in\mathbb{R}^3:|x|<|z|^2,|y|<|z|,0<z<1\}$ and $f:G\to\mathbb{R},f(x,y,z)=2x+2y+z^3$. Calculate the surface of the graph of f. We recently got introduced to Stokes' ...
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2answers
239 views

How many charts are needed to cover a 2-torus?

Can you please answer this question with explanation ? I just learned about the charts needed to cover 1 and 2 spheres but got confused for the case of torus. It would be great if you guys could help. ...
2
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1answer
78 views

Tangent vector to a curve on a manifold

If one has a curve $\sigma : (-1,1) \rightarrow M$, where $M$ is a smooth manifold, the tangent vector in $\sigma(0)$ is usually defined as $$ \sigma'(0) (f) = \dfrac{d f \circ \sigma}{dt} \Big|_0,$$ ...
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1answer
42 views

Proving that charts are related

Let $A$ be an atlas on the set $M$ and let $ x: U \to x(U) $ and $ y : V \to y(V )$ be bijections from subsets $U, V \subset M$ to open sets $x(U), y(V ) \subset \mathbb{R^n} $. Show that if the ...
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0answers
67 views

How to prove that the composite function is smooth

Let $f:M \to N$ and $g:N \to K$ be smooth functions, where $M,N$ and $K$ are smooth manifolds. How to prove that the composite function $g \circ f$ is smooth, noting that the Chain Rule only applies ...
3
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1answer
57 views

How to verify that this is a submanifold

Let $ g: \mathbb{R}^2 \to \mathbb{R}^2 $ , $ g (x, y) = (x^2-y^2, y) $ be a differentiable map. Let $ r $ the line passing through $(1, 0) $ parallel to the $ y-$axis. Prove that $ g^{-1}(r) $ is a ...
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1answer
85 views

Manifold is $2$nd countable iff it has a countable atlas

I am trying to prove that if a smooth manifold has a countable atlas than it has a countable basis. If I have $(U_n, \varphi_n)$ a countable atlas, how can I find a basis of the topology? If I take ...
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2answers
128 views

Recommendation on studying Smooth Manifold & Differential Geometry and related subjects.

My major was physics, but i'm changing my major to mathematics this year. I took 2 years off to study mathematics myself and now i'm going back in this year. Here's the list of books i have studied ...
3
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1answer
106 views

prove that a map $g: M \to N$ is smooth

Let $A$ be an $n \times n$ orthogonal matrix (that is, the columns of $A$ are perpendicular to one another and have length $1$). Then multiplication by $A$ defines a map $f:\mathbb{R}^n \to ...
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1answer
37 views

Proof of Morse inequalities?

Can you think of a proof of Morse inequalities without using the Morse cohom. $\cong$ sing.cohom or Witten's approach? http://en.wikipedia.org/wiki/Morse_theory#The_Morse_inequalities Any references ...
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1answer
40 views

integration of differential forms on covering space

Let $M_1,M_2$ be $n$-dimensional oriented manifolds. Let $f: M_1\longrightarrow M_2$ be an orientation-preserving diffeomorphism. Then for any $\omega\in \Omega^n_c(M_2)$ we have(page 85 of {Madsen: ...
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1answer
62 views

What is the best (in terms of effectively building understanding) direction from which to approach manifolds?

Thedore Frankel's book The Geometry of Physics presents Manifolds right away in Chapter 1 in the following manner: Introduce the Euclidean space $\mathbb{R}^N$ only as "the most important manifold". ...
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1answer
21 views

Atlases and Transition Maps

What is the difference between open sets and open balls? Definitions of atlases for manifolds do not seem to specify any difference.
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1answer
29 views

Vanishing of 1-form

If $\theta \in \frak{X}^* \mathrm{(M)}$ and $\theta (X) = 0$ $\forall X \in \frak{X} \mathrm{(M)}$ then $\theta = 0$. How do I prove this statement? Consider a manifold $M$ with chart $x^1, \dots, ...
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64 views

An (extended) semimetric on surfaces

Given a smooth surface $S \subseteq \mathbb{R}^3$, like the surface of sphere, we can define the following extended semimetric $d : S^2 \to [0, \infty]$, where $$ d(x,y) = \inf\{\lVert x - p\rVert + ...
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1answer
39 views

Significance of bump function in the proof.

In the book Semi-Riemannian Geometry with applications to Relativity by Barrett O'Neill, in Chapter 2 (Tensors), he stated that: However, I don't get why he had to use a bump function in the proof. ...
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1answer
67 views

Topological Manifolds & Covers

This problem is from John Lee's "Introduction to Smooth Manifolds" 1-4. Let M be a topological manifold, and let U be an open cover of M . (a) Assuming that each set in U intersects only finitely ...
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1answer
104 views

$V$ vector field, $\omega$ one-form, $V(\omega(V))$=?

(1-forms) Let $X$ be a manifold and $\omega \in \Omega^1(X)$ be a smooth 1-form, and $V, W \in V^{\infty}(X)$ smooth vector fields on $X$. Then, $\omega(V ), \omega(W ) \in C^{\infty}(X)$ are ...
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80 views

Lie Bracket and vector fields

Could you please help me solve it? Let X and Y be smooth vector fields on $\mathbb{R}^n$. Suppose that $[X,Y]=0$ (Lie bracket). Show that around each point there exists local diffeomorphism f such ...
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1answer
50 views

Let $f:\mathbb{R^n} \to \mathbb{R^m}$, if $f$ is a linear transformation, prove that $Df(a)=f$.

Let $f:\mathbb{R^n} \to \mathbb{R^m}$, if $f$ is a linear transformation, prove that $Df(a)=f$. My try : By definition of derivative of a function $f:\mathbb{R^n} \to \mathbb{R^m}$ , If I know ...
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1answer
27 views

Is a one-form a derivation on $C^ \infty$?

I know that a vector field is a derivation on $C^ \infty$, meaning that it is R-linear and Leibnizian. Is it the same case for one-forms?
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41 views

Cone is a submanifold iff it is a vector subspace

Could you tell me how to prove the following? Let $\emptyset \neq M \subset \mathbb{R}^n $ be a cone ($\forall x \in M, t \in \mathbb{R} : tx \in M $ ), $0 \le d \le n.$ Prove that $M$ is a $d$ ...
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0answers
32 views

How to find a curve inside a non-convex

I want to connect two points in a space within the space. If the space is convex, I can simple draw a line between them. But how about a non-convex space. How can I find a curve connecting these two ...
4
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1answer
128 views

Geometric interpretation of Supersymmetry

Is there a geometric interpretation of supersymmetry? I.e., if one has a manifold $\mathcal {M} $, and there are $\mathcal {N} $ SUSY generators, then is there a geometric interpretation of the SUSY ...
2
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0answers
71 views

Local dimension of graph embedding

I am trying to find a way to characterize the dimension of the smallest space into which a (neighbourhood of) a graph $\Gamma = (V, E)$ may be embedded. Although in the end my goal is to identify what ...
6
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1answer
139 views

Lower bound for the size of an atlas

This question came up in a graduate-level class on differential topology I'm currently taking; the instructor couldn't come up with an answer off the top of her head and while I'm very new to the ...
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0answers
54 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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2answers
30 views

Multilinearity of the exterior derivative of a one-form.

I wish to show that the exterior derivative $d \theta$ of a one-form $\theta$ is $\frak{F} \mathrm{(M)}$-multilinear, therefore, a tensor. Let $X, Y, V, W \in \frak{X} \mathrm{(M)}$ and $f, g, h, k ...
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1answer
137 views

Is this an abuse of notation?

Here is a proof says that the differential of Gauss map is self-adjoint. But I seems there is an abuse of notation at (1) in it. Since $dN_p$ is linear, it suffices to verify that $\langle ...
2
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1answer
76 views

How to decide if a given set is a manifold

Could you tell me how to decide if a certain set is a manifold? I know there already is a similar question here, but there we have fairly "visualizable" sets: a hemisphere and a square. What in ...
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1answer
45 views

Vanishing of a covector (1-form) and a vector field

a) A one-form $\theta$ is zero if and only if $\theta X = 0$ for all $X$ in the set $\frak{X} \mathrm{(M)}$ of all smooth vector fields on a manifold $M$. b) A vector field $X$ is zero if and only ...
0
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1answer
76 views

Path or One-Dimensional Manifold

The terminology of curve, path and one dimensional manifold is always in the textbooks about topology, differential manifold and riemannian geometry. The definition of path and one dimensional ...