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

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35 views

Proof of $G\rightarrow G/H$ is a Principal H bundle

Let $G$ be a Lie group and let $H$ be a closed subgroup (not necessarily normal). Then $G$ is a principal $H$-bundle over the (left) coset space $G/H$. I could proof that the fibers are all ...
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2answers
44 views

Definition of critical point

Let $f:M→N$ be a smooth function between two smooth manifolds. Then $p\in M$ is a critical point if $df_p$ is not surjective. I feel very confused about this definition, even in the case where ...
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1answer
52 views

Differential equation on a manifold

I want to solve this problem : M is a manifold. Let $t\mapsto \gamma(t)$ be an integral curve of a vector field X on M. Suppose there exists $t_0$ such that $\gamma'(t_0)=0$. Prove that ...
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0answers
28 views

Is $H^1(\Omega, S^2)$ a Hilbert manifold?

I'm considering the topology of the function space $H^1(\Omega, \mathbb{S}^2)$, where $\Omega \subset \mathbb{R}^3$ is a bounded domain. Obivously it is not a vector space, but is it a Hilbert ...
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1answer
51 views

Integral over vector field

Can someone help me with this one: Let $X$ be a vector field on $R^{3}$ such that $X(x)=x$, for each $x\in S^{2}$. Calculate $$\int\limits_{\overline{B^3(1)}} \left(\frac{\partial X_1}{\partial x_1} ...
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1answer
42 views

Explaining what a symbol $W^{\vee}$ means

I've found here http://math.stanford.edu/~conrad/diffgeomPage/handouts.html a very interesting paper on Stokes theorem for manifolds with corners. So I've decided to read a paper on manifolds with ...
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1answer
53 views

Associated bundles: isomorphism between spaces of differential forms.

I think this will be an easy question for numerous people. Let $\pi:P\rightarrow M$ be a principal bundle and $\rho:G\rightarrow GL(V)$ a representation. The space of $k$ forms on $M$ with values in ...
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0answers
20 views

Gauge transformation laws, proof in Kobayashi & Nomizu Foundations of Differential geometry

I have two questions about this proof found in K&N's Foundations of Differential Geometry. 1) Can someone please explain how they deduce ...
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1answer
29 views

Derivative of a function on a manifold

I want to show that: Given $f,g \in C^\infty(M)$ defined in a differential manifold of dimension $n$ and $a \in M$, we have $$(dfg)_a=f(a)(dg)_a+g(a)(df)_a,$$ using the following proposition: ...
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0answers
27 views

Infinitesimal Generator of A One Parameter Group

This is a small problem which drives me crazy. Let $\varphi(x,y,t)=(F_1(x,y,t),F_2(x,y,t))$ be a one paramter transformation group on $\mathbb{R}^2$. Let ...
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0answers
49 views

Comparing normal bundles of embedded submanifolds and their sections.

Let $M,M'\subseteq \mathbb{R^n}$ two compact embedded submanifolds, which are abstractly diffeomorphic. Tangent and normal bundle of the two submanifolds inherit a metric from $\mathbb{R^n}$. By the ...
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1answer
60 views

Examples of important manifolds that are direct products of non-trivial manifolds

In this question, I asked for interesting / non-trivial examples of smooth connected closed manifolds that happen to be direct products or involve direct products, especially orientable manifolds. In ...
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3answers
215 views

Examples of interesting / non-trivial manifolds that are direct products

What are interesting / non-trivial examples of smooth connected closed manifolds that are direct products or involve direct products? I am especially interested in orientable manifolds. Say, an ...
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2answers
66 views

Gauge fields and restrictions of the connection one form

I am working through some lecture notes on principal bundles and am stuck on the proof of a certain proposition. In the following, $\pi:P\rightarrow M$ is a principal bundle, $\omega$ is the ...
2
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1answer
43 views

Connected Sum of Surfaces

I am trying to prove that the connected sum of surfaces is a surface. My definition of surface is: A topological space locally homeomorphic to $\mathbb{R}^2$, second countable, Hausdorff and ...
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0answers
43 views

Clarification of vector valued forms (sections and tensor products)

I am seeking a bit of clarification on vector valued forms. Intuitively, a vector valued differential form is a differential form on $M$ whose values lie in some vector space. More accurately, Let ...
2
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1answer
51 views

The curvature of the connection one form - misunderstanding

Let $(P,M,G,\pi,\cdot)$ be a principal bundle. Let $\omega$ be the connection one form for a connection $H\subset TP$. Let $X,Y$ be smooth vector fields on $P$. Then the curvature $\Omega$ of the ...
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2answers
60 views

Definition of differential on manifolds

I'm studying some differential Geometry at the moment and I'm getting a bit stuck with the definition of the differential. It's defined as follows \begin{array}{cl} \phi_{\star,m} : T_{m}M ...
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0answers
28 views

Intuition about twisted homology

I will need to do quite some reading on twisted invariants this year, such as twistings of Reidemeister Torsion or Alexander Polynomials. I already had some insights into said (twisted) invariants, ...
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1answer
54 views

Prove that an atlas is $C^{\infty}$

The sphere $S^2$ can be covered by the following $6$ subsets (hemispheres) $$ O_i = \{(x^1, x^2, x^3) \in \mathbb{R}^3 | x^i > 0, i = 1, 2, 3\}$$ Each of these subsets can be mapped by the unit ...
3
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1answer
36 views

Higher homology groups relative a lower dimensional subspace

One often works with reduced homology, which (in the case of say, smplicial homology) is defined as the homology relative a point. Now at every grade except zero, the reduced homology objects are ...
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0answers
17 views

How two disjoint solid 2-tori linked?

I have 3-manifold which is a union of two disjoint solid 2-tori, How I can decide if they are meet or no? Also if they are meet how I can know the way they are linked?? Thanks in advance
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0answers
55 views

$F$-related vector fields

I'm having a bit of difficulty understanding $F$-related vector fields. I think I understand conceptually what's going on (taking a vector field on a manifold $M$ and getting a smooth vector field on ...
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2answers
80 views

Lie Groups/Exponential map identity

I have come across this identity a few times and I have absolutely no idea why it holds. $g^{-1}\exp(tX)g=\exp(t(\text{ad}_{g^{-1}}X))$ Would any one be able to explain exactly why this holds or ...
0
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1answer
31 views

Step in a proof about alternating operators

The theorem is that if $f$ is a $k$-linear function on a vector space $V$, then the $k$-linear function $Af$ is alternating. $\def\sgn{\operatorname{sgn}}Af=\sum (\sgn \sigma)\sigma f$ Proof: ...
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1answer
32 views

Definition of connection on vector bundle

A connection on a vector bundle $E$ is a map $ D:\Gamma(E)\rightarrow \Gamma(T^*(M)\otimes E)$ satisfying 1) For any $s_1,s_2\in \Gamma(E)$, $D(s_1+s_2)=Ds_1+Ds_2$ 2) For $s\in \Gamma(E)$ and ...
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0answers
16 views

Inverse Mapping Theorem and open mapping

Apparently, one consequence of the Inverse Mapping Theorem is that if $f: U \rightarrow \mathbb R^n$ is a continuously differentiable mapping defined in an open subset $U \subset \mathbb R^n$ such ...
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0answers
31 views

Fibers of polynomial as submanifold

Let $f\in \mathbb{R}[X_1,...,X_n]$ be a homogeneous polynomial of degree $d$. Let $F_{a}=\{(x_1,...,x_n)\in\mathbb{R}^n : f(x_1,...,x_n)=a\}$. For which $a$ $F_a$ is a submanifold in $\mathbb{R}^n$ ? ...
0
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1answer
51 views

Tangent cone and tangent space

Let $M$ be a set in $\mathbb{R}^n$ such that $M$ is locally a graph of some differentiable function (not necessarily $C^1$ ). Let $p\in \bar M$. We can define a tangent cone $C(M,p)$ as a set of all ...
1
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1answer
36 views

What are properties of dynamical systems in non-integer dimension spaces?

A 1D dynamical system (R1) exhibits convergence to a fixed point, or escapes to infinity. A 2D dynamical system (R3) can produce oscillations, spiral-shaped trajectories, etc. A 3D dynamical system ...
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0answers
32 views

A question on Grassmannian manifolds and universal line bundles

Problem. Fix an $ n \in \mathbb{N} $. Is it true that there exists a $ k \in \mathbb{N} $ such that every smooth complex line bundle over a smooth $ n $-dimensional real manifold is the pullback of ...
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1answer
73 views

Proving existence of local diffeomorphism

Consider the setup from here: Do these vector fields span an integrable distribution? For any pair of points $p, q \in U$, show that there is a local diffeomorphism $F: U(p) \to U(q)$, such that ...
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1answer
72 views

Elementary tensors [duplicate]

I need to determine whether the following function is tensor on $\Bbb R^4$ and express it in terms of elementary tensors. Can someone please help me with it? I do not know what elementary tensor means ...
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0answers
18 views

Hausdorff Dimension of a k-dimensional submanifold, hence $L^n(M) = 0$

If $M \in P(R^n)$ is a k-dimensional submanifold, show that the Hausdorff-Dimension of M is k. Also, is there a quicker/easier way to show, that $L^n(M) = 0$, where L is the Lebesgue-Measurement.
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1answer
28 views

Is the unit square a submanifold/manifold?

In my course we have just been introduced to and will only be dealing with regularly embedded submanifolds. Let $M = [0,1]\times [0,1] \subset \mathbb{R}^2$. I don't think it's a submanifold. If it ...
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1answer
23 views

Construct a diffeomorphism $[a,b] \rightarrow [c,d]$ with slope 1 at $a$ and $b$

I'm trying a problem from An Introduction to Chaotic Dynamical Systems regarding bump functions. At this point, we have successfully constructed, for any $\alpha < a < b < \beta$ a bump ...
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2answers
63 views

A reference for a result by A. Casson

I was reading this article about the disproof of Triangulation conjecture: it says that A. Casson disproved this conjecture in dimension 4 in the '80s In 1982, Michael Freedman, then at the ...
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1answer
53 views

$M=\{(x,y,z) \in \mathbb{R}^3 : xyz=C\}$ is a manifold $\Rightarrow C \neq 0$

I'm having some troubles on showing that $M=\{(x,y,z) \in \mathbb{R}^3 : xyz=C\}$ is a manifold $\Leftrightarrow C \neq 0$ I have already proved $\Leftarrow$ but I can't see how to prove ...
3
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0answers
65 views

Homotopically equivalent manifolds and product with $\mathbb{R}$

I know that some manifolds which are homotopically equivalent become homeomorphic after taking the product with $\mathbb{R}$, e.g. $\mathbb{T}^{2}$ minus a point and $\mathbb{S}^{2}$ minus three ...
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2answers
59 views

Can manifolds be defined over fields other than $\mathbb{R}$ and $\mathbb{C}$?

The question is one in the title. By other fields, I mean fields like $\mathbb{Q}$. But I have never seen manifolds defined over fields other than $\mathbb{R}$ and $\mathbb{C}$. Even if one defines, ...
2
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1answer
46 views

Showing that $f: \mathbb{R}^2 \to \mathbb{R}^n$ can't be a homeomorphism for $n>2$

Without using the Invariance of Domain result, I want to show that such an f cannot exist. Here is what I did: Assume that there is an $f: \mathbb{R}^2 \to \mathbb{R}^n \ , n>2$ that is ...
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1answer
27 views

Derivative of a function - how to compute for those examples

I'm taking a Diferential Manifolds course and I don't understand how to compute $DF_a$ in order to apply the following theorem: Let $F:U \rightarrow \mathbb{R}^m$ be a $C^\infty$ function on an ...
2
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1answer
21 views

matrix Lie group embedding as a manifold

Given a Lie group of matrices, and suppose for simplicity that it is globally generated through exponential map from its Lie algebra on a element. Is there a canonical way to embed it into ...
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1answer
28 views

Show that the covering space of a smooth manifold is a smooth manifold.

I indeed found this question Is a covering space of a manifold always a manifold. However I do not know the concepts here used. As far as I know I just need to present a suitable atlas for the ...
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0answers
38 views

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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0answers
32 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
34 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 ...
3
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1answer
38 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 ...
0
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0answers
17 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
53 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. ...