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

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2answers
22 views

Any neighbourhood of a point $x$ in a manifold $X$ ($\dim X \geq 2$) has a subneighbourhood $V$ of $x$ such that $V \setminus \{x\}$ is connected

What I want to show for this is that $V \setminus \{x\}$ is homeomorphic to some punctured ball, $B_m \setminus \{p\}$ (where $B_m$, $p \in \Bbb R^m$). And then since $B_m \setminus \{p\}$ is ...
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1answer
43 views

Curves and tangent vectors in a manifold setting

Consider the following definition: ($M$ denotes a manifold structure, $U$ are subsets of the manifold and $\phi$ the transition functions) Def: A smooth curve in $M$ is a map $\gamma: I \rightarrow ...
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1answer
88 views

Manifolds, coordinate systems, books

Which books, say Lee's Introduction to Smooth Manifolds or Munkres' Analysis on Manifolds explains how the theory of a differentiable manifolds can be used to solve a problem that is expressed in a ...
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1answer
62 views

Extension Lemma for Smooth maps (Lee vs. Lee)

I've been reading Jeffrey Lee's, Manifolds and Differential Geometry and John Lee's, Introduction to smooth manifolds. In the first book (here, in page 31), after introducing partition of unity, ...
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1answer
27 views

Checking alternating tensors

How do I check that $$f(x,y)=x_1y_2-x_2y_1+x_1y_1$$ is an alternating tensor? I did check that f is a tensor, but how do I know if it is alternating by direct computation? Thanks in advance!
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1answer
34 views

Examples of mod 2 degree of f.

I'm reading Milnor's Topology from the Differentiable Viewpoint and I've just finished the chapter about degree mod 2. Since every two smooth homotopic maps $f,g$ must have the same degree mod 2 ...
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0answers
27 views

Doubt in an expression of the vector field

I know that $$\{\dfrac{\partial}{\partial x_i}:i=1(1)n\}$$ is a basis of the $n$ dimensional tangent space $T_p(M)$ [Vector space over $\mathbb R$] at the point $p$ on $M.$ Again I came to know ...
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1answer
31 views

Exterior derivatives involving representations

I have two questions regarding the exterior derivative of vector valued forms when representations are involved: Suppose $V$ is a vector space, $M$ a smooth manifold and $\omega$ is a $V$ valued ...
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1answer
48 views

Tangent space of tangent vector

Let $M$ be a smooth manifold. There's a (split) short exact sequence $$0\to T_aM\to T_v(TM)\stackrel {D_v}{\to} T_aM \to 0,$$ where $v\in TM$ and $a=\pi(v)\in M$. I'm trying to understand what this ...
4
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2answers
83 views

Principal bundles, connection forms and fundamental vector fields

Suppose $\pi:P\rightarrow M$ is a principal bundle, $\omega\in \Omega^1(P;\mathfrak{g})$ is the connection one form and $\sigma(\cdot)$ is the fundamental vector field associated to some vector field ...
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1answer
63 views

A question about the Möbius Strip and the Projective Plane

I know that both the Möbius Strip and the Projective Plane are both 2-manifolds. I try to prove that they are locally homeomorphic to $\mathbb{R}^2$ and Hausdorff. It seems easy to see that the ...
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0answers
24 views

Local submersion theorem and $O_{3}(\mathbb{R})$

I was attempting to follow the proof of the local submersion theorem given in Differential Topology by Guillemin & Pollack in the case that $X = O_{3}(\mathbb{R})$ and $f(A) = AA^{T}$. I worked ...
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0answers
38 views

A kind of uniqueness for the double of a manifold

Let $M$ and $N$ be two manifolds with the same boundary. If their doubles $D(M)$ and $D(N)$ are diffeomorphic, are $M$ and $N$ diffeomorphic?
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0answers
60 views

Munkres' Analysis on Manifolds and Differential Geometry

Will Munkres' Analysis on Manifolds prepare me for a text like John Lee's Introduction to Topological Manifolds and his Introduction to Smooth Manifolds text? Would one be able to successfully tackle ...
2
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1answer
41 views

Concrete example of zero section

I just learnt tangent bundle and I want to get some intuition about zero section (and sections in general). I'm even not clear about what the zero vector is in a tangent space--e.g. just consider ...
4
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1answer
52 views

Ways of thinking about vector-valued differential forms

I am trying to get a better intuition of vector-valued differential forms. Let $V$ be a vector space and $M$ a smooth manifold. Consider the space $\Omega^k(M;V)=\Gamma((M\times V)\otimes ...
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2answers
102 views

What are some good sources to learn about real analytic manifolds?

Asking this question as someone with a graduate student level understanding of smooth differential/Riemannian geometry (May be a bit more that Riemannian Geometry by Do Carmo). I am trying to upgrade ...
2
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1answer
70 views

On the definition of critical point

Let $f:\mathbb{R}^n\to \mathbb{R}^m$ be a smooth function (or in general between two smooth manifolds). Then $p\in \mathbb{R}^n$ is a critical point if $df_p$ is not surjective. I feel confused about ...
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1answer
44 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 ...
2
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2answers
49 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 ...
6
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1answer
60 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
30 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
52 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
45 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 ...
3
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1answer
55 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
25 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
36 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: ...
2
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0answers
30 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 ...
1
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0answers
51 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 ...
3
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1answer
62 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
229 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
68 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
46 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 ...
2
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0answers
49 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
55 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
64 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
33 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, ...
3
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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
38 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
18 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
64 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 ...
4
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2answers
90 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
32 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
35 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
22 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
32 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
74 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 ...
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1answer
37 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
37 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 ...
3
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1answer
81 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 ...