For questions about smooth manifolds.

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12 views

Good reference for The Differntiable Slice Theorem

I am looking for a book that will give me a good proof of The Differentiable Slice Theorem - Suppose a compact Lie group $G$ acts smoothly on a manifold $M$. Then every orbit has a $G$-invarient ...
2
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1answer
57 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 ...
2
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2answers
116 views
+100

Proof in Hamilton: Divergence theorem for differential forms?

For a vector field $X\in\Gamma(TM)$ on a closed Riemannian manifold $(M,g)$ we have \begin{align*} \int_M\text{div}X\;\mu=0, \end{align*} where \begin{align*} ...
2
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0answers
25 views

$C^1$ versus smooth submanifolds

Is the graph of $f(x)=|x|\,x$ (or any $C^1$ function that is not $C^\infty)$ a smooth embedded submanifold of $\mathbb{R}^2$ with its standard differential structure? I apologize if this is too ...
1
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1answer
66 views
+250

Gluing submanifolds along their common boundary

Let $M$ be a smooth $m$-manifold and $N_1$, $N_2$ smooth embedded $k$-submanifolds such that $N_1\cap N_2=\partial N_1=\partial N_2$, for each $x\in N_1\cap N_2$, $T_x N_1=T_x N_2$, and for $x\in ...
3
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3answers
48 views

Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...
2
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1answer
49 views

Relationship between differential forms and coordinates

For the purpose of this question let us restrict our considerations to smooth $3$-manifolds. So the manifold $M$ we consider here is endowed with smooth coordinate charts $(x,y,z)$. What I have ...
3
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1answer
42 views

Embedded Submanifolds Have a Unique Smooth Structure

Let $M$ be a smooth manifold. An embedded submanifold of $M$ is a subset $S$ of $M$ such that $S$ is a topological manifold under the subspace topology induced by $M$, endowed with a smooth structure ...
6
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0answers
56 views

Differentiation on Manifolds Basics

I'm having some real trouble comprehending integral curves and Lie derivatives on a Manifold. I will write out my understanding and ask the questions below. For a vector field $X$ on smooth manifold ...
5
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0answers
48 views

Intuitive Aproach of Dolbeault Cohomology

I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. All suggestions are welcome.
3
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1answer
73 views

Intuitive Approach to de Rham Cohomology

The intuition behind homology may be summarized in a sentence: to find objects without boundary which are not the boundary of an object. This has geometric meaning and explains the algebraic boundary ...
3
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1answer
35 views

Proving that something is a manifold from the definition

Consider a set $$M = \{ (s\cos t, s\sin t, t) \colon s,t\in \mathbb{R}\}\subset \mathbb{R}^3.$$ I am asked to show from the definition that $M$ is a 2-dimensional submanifold of $\mathbb{R}^3$ ...
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2answers
51 views

If $f : M\rightarrow N$ be immersion then $f_*$, derivative of $f$ is an immersion. [closed]

Suppose that $f : M\rightarrow N$ be immersion. Prove that $f_*$, derivative of $f$, is immersion too?
4
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2answers
111 views

Elementary proof of the fact that any orientable 3-manifold is parallelizable

A parallelizable manifold $M$ is a smooth manifold such that there exist smooth vector fields $V_1,...,V_n$ where $n$ is the dimension of $M$, such that at any point $p\in M$, the tangent vectors ...
1
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0answers
14 views

Introduction to Euler structures

I am looking for a basic text on Euler structures, in particular smooth Euler structures, and the relation to combinatorial Euler structures; It is known that given a combinatorial Euler structure on ...
1
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1answer
28 views

Find an explicit atlas for this submanifold of $\mathbb{R}^4$

I'm having a hard time coming up with atlases for manifolds. I can prove using the implicit function theorem that $M = \{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4:x_1^2+x_2^2=x_3^2+x_4^2=1\}$ is a ...
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2answers
40 views

Give an explicit atlas for the manifold $\mathbb{R}P^3$ which is defined as the quotient of the three-sphere by the antipodal mapping.

I'd like to know if there is an explicit atlas for the manifold $\mathbb{R}P^3$ which is defined as the quotient of the three-sphere by the antipodal mapping. Thanks.
0
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1answer
20 views

What is the significance of incompatible coordinate charts for a manifold?

For reference, here is my definition of a "manifold". A $\,C^\infty$ manifold is a topological manifold together with all the admissible charts of some $C^\infty$ atlas. When considering the ...
4
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1answer
73 views

Computation of Pushforward and Pullback

After reading about the pushforward and pullback, I don't really have a concrete grasp of them, so I think these simple questions might clear things up for me; I appreciate any hints or solutions. ...
4
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1answer
122 views

Parametrised vs Regular Surfaces

Two types of surfaces in $\mathbb{R}^3$ are usually studied in introductory books on differential geometry: Parametrised or immersed surface: Is an immersion $F:U\rightarrow\mathbb{R}^3$ from an ...
0
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0answers
22 views

regular submanifolds of a connected manifold

let M be a connected smooth manifold and f is a smooth function from M to itself.and f has the property that f(f(X))=f.how to show that f(M) is a regular submanifold of M?
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0answers
43 views

Local expression of a differential form

During the course, we have defined differential forms as maps $$ \omega : D(M) \times \cdots \times D(M) \rightarrow \mathcal{C}^\infty(M), $$ where $D(M)$ are the global derivates of the manifold ...
2
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1answer
91 views

Spivak Calculus on Manifolds, Theorem 5-2

In the proof Theorem 5-2 of Spivak Calculus on Mannifolds how is \begin{align*} V_2\cap M=\{f(a):(a,0)\in V_1\}? \end{align*} (That $\{f(a):(a,0)\in V_1\}=\{g(a,0):(a,0)\in V_1\}$ is clear.) Edit: ...
2
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2answers
61 views

Equivalent definitions of a surface

do Carmo Differential Geometry of Curves and Surfaces defines a regular surface as per the below post. Lee Introduction to Smooth Manifolds defines an embedded or regular surface to be an embedded or ...
2
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1answer
42 views

Topological covering + local diffeomorphism gives smooth covering

I got stuck at some point while working on this part of an exercise from Lee's Introduction to Smooth Manifolds, 2nd edition. The part which I am stuck on is to prove (one of the directions of ...
4
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2answers
123 views

Why is first fundamental form considered intrinsic

I am reading Kuhnel's differential geometry book, and in chapter 4, it says that "intrinsic geometry of a surface" can be considered to be things that can be determined solely from the first ...
1
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1answer
20 views

Refinement of an Open Cover

This seems rather simple, but just curious about the following definition (pulled from Lee, but definitely standard): Given an open cover $\mathcal{U}$ of $X$, another open cover $\mathcal{V}$ ...
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2answers
36 views

$L_X( \omega (Y_1,\ldots, Y_n)) = (L_X \omega) (Y_1, \ldots, Y_n) + \sum_{i = 1}^n \omega (Y_1, \ldots, Y_{i-1}, L_X Y_i, \ldots, Y_n)$

Here $\omega$ is a smooth form on a manifold, $X, Y_1, \ldots, Y_p$ are smooth vector fields, and $L_X$ represents the Lie derivative. I am having trouble proving this standard identity. All the ...
0
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0answers
42 views

Stereographic Projection on $S^n$?

I'm studying differentiable manifolds and I'm trying to work out the details of the stereographic atlas for $S^n$. I'm trying to deduce the expressions for the stereographic projection. I did a ...
3
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2answers
91 views

Applying the Frobenius theorem to a decomposable 2-form

So I have the following problem: Suppose $\omega=\phi \wedge \theta$ is a closed decomposable 2-form on $M$ a manifold (decomposable just means it can be written as a wedge of 1-forms). Suppose $p\in ...
4
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1answer
47 views

How is defined the notion of $C^1$-close submanifolds?

The question is already in the title. Reading some papers, I have find statements like the following one, with no reminder about the notion of $C^1$-closedness, and without references for further ...
12
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1answer
151 views

Intuition for Exotic $\mathbb R^4$'s

Today one of my professors told me that $\mathbb R^4$ admits uncountably many non-diffeomorphic differential structures. When I asked him whether there's an intuitive reason to expect a result like ...
3
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1answer
46 views

mutually transverse embedded submanifolds, natural bundle surjections, direct sum, isomorphism

Let $N$ be a manifold and let $M_1, \dots, M_n \hookrightarrow N$ be mutually transverse embedded submanifolds, so $M = \cap M_i$ is an embedded submanifold of $N$ with $\text{T}_m(M) = \cap T_m(M_i)$ ...
3
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0answers
93 views

Fundamental group of connected sum for non-orientable manifolds

For orientable $n$-manifolds, $n\ge3$, the fundamental group of connected sum is free product of fundamental groups: $\pi_1(M\#N)=\pi_1(M)*\pi_1(N)$. As far as I understand, for non-orientable ...
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1answer
40 views

Surjection of the fundamental group of a manifold onto a free group induces a map onto a wedge of circles, why?

Why does a surjective map $\pi_1(M)\twoheadrightarrow F$ of the fundamental group of a manifold $M$ onto a free group $F$ over $n$ generators induce a continuous map $M\twoheadrightarrow\bigvee^n S^1$ ...
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0answers
29 views

Sheaf-theoretic approach to Morse functions?

It is known that one can define a smooth structure on a manifold using a sheaf-theoretic formulation via defining the algebra of the (a fortiori) smooth functions on it (which satisfies the usual ...
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1answer
57 views

Properly discontinuous action: equivalent definitions

Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = ...
1
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1answer
40 views

Two definitions of embedded submanifold

Let $N$ be a smooth manifold. One possible definition (I believe) for an embedded submanifold of $N$ is some $M \subset N$ that is a (smooth) manifold such that the inclusion $i : M \hookrightarrow N$ ...
1
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1answer
136 views

The element of Volume of the Sphere and two formulas

Let $S^{n-1}$ be the unit sphere with the inner product $<.,.>$ that inherits from $\mathbb{R}^n$ and $V\in S^{n-1}$. Let $\{e_ i \}_{ i=1}^n $ be an orthonormal frame and let ...
0
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1answer
87 views

Example closed 1-form on $\mathbb{R}^3 -\{0\}$

It's maybe a silly question but I was wondering if there exists a closed 1-form $\alpha$ on the manifold $\mathbb{R}^3 -\{0\}$ of the form $$\frac{1}{x^2+y^2+z^2}(adx+bdy+cdz)$$ with $a, b$ and $c$ ...
5
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4answers
78 views

group operations are smooth in $\text{SL}(n, \mathbb{R})$

I am told the following reason as to why group operations of multiplication and inversion are smooth on $\text{SL}(n, \mathbb{R})$. Multiplication is smooth because the matrix entries of a product ...
2
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1answer
54 views

Differentiable manifolds that allow isometric transition maps.

What is the class of differentiable n-dimensional manifolds that allow a differential structure, in which all transition maps are isometric? Note that isometric must be overlapping pieces of charts ...
3
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1answer
37 views

defining $C^\infty$ structure on finite-dim vector space, homeomeomorphism to tangent bundle, such that independent of choice of bases

If $V$ is a finite dimensional vector space over $\mathbb{R}$, how would I go about defining a $C^\infty$ structure on $V$ and a homeomorphism from $V \times V$ to $TV$ which is independent of bases? ...
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0answers
20 views

An example of regular singular points

I was reading a book on differential geometry and after the intro to the concepts of regular singular points I came across an example under it: The set $M:=\{(x^2,y^2,z^2,yz,zx,xy)|x,y,z\in ...
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1answer
28 views

Quick question: Map being smooth vs Graph being submanifold of the product space [closed]

Is $f:X\rightarrow Y$ smooth if and only if the graph $\Gamma_f$ is a closed submanifold of $X\times Y$? Thank you very much.
7
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3answers
308 views

parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
1
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1answer
58 views

Representation of $n$ form and $n-1$ form in local coordinates

Let $M$ denote a smooth $n$-dimensional manifold. (a) Let $\phi$ denote a smooth $n$ form which is nowhere zero. Show that every $x_{0} \in M$ has a neighborhood on which we can find smooth local ...
2
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1answer
58 views

identity map is not diffeomorphism, $x^3$ is a diffeomorphism [closed]

Consider the real line $\mathbb{R}$ the two following differentiable structures: 1) $(\mathbb{R}, f_1)$ where $f_1(x) = x$. 2) $(\mathbb{R}, f_2)$, where $f_2(x) = x^3$. How do I demonstrate that: ...
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3answers
241 views

“Drawable” Examples of Vector Bundles

I'm looking for examples of vector bundles that can be easily drawn or "illustrated" on a whiteboard for a talk I am giving. I know of a couple simple examples that I could use: When our base ...
2
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1answer
43 views

Smooth function on intersection is the difference of two smooth functions

I am trying to understand a proof from Loring W. Tu's An introduction to Manifolds. In order to prove Proposition 26.2, The author must show that if $\{U, V\}$ is a open cover of a manifold $M$ and ...