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

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2
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1answer
72 views

Diagonal Inclusion Map of a manifold $X$

This question actually comes from the question I asked before: Derivative map of the diagonal inclusion map on manifolds And I repeat it as follows: Let $f: X\longrightarrow X\times X$ be the ...
11
votes
1answer
209 views

The “Easiest” non-smoothable manifold

In 1960, Kervaire found the first example of a PL-manifold which does not admit a smooth structure. Since then, I understand that there are many examples of non-smoothable manifolds that can be built. ...
4
votes
1answer
110 views

Horn and spindle tori

I was trying to prove that the horn torus and the spindle torus are not manifolds by definition(locally diffeomorphic to some Euclidean space.). I have no idea how to do this, but I attempted it in ...
6
votes
1answer
179 views

Derivative map of the diagonal inclusion map on manifolds

I was trying to work through a problem(#10 of $\S$1.2) in Guillemin and Pollack's book $\textit{differential topology. }$ The problem is given as follows. Let $f: X\longrightarrow X\times X$ be ...
2
votes
1answer
90 views

Finding a direct basis for tangent space of piece with boundary of an oriented manifold.

I have the following definition (from Hubbard's vector calculus book) for an oriented boundary of piece with boundary of an oriented manifold: Let $M$ be a $k$ dimensional manifold oriented by ...
1
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1answer
92 views

$[M,\mathbb CP^\infty]=[M,\mathbb CP^2]$ where $M$ is a smooth closed orientable 3-manifold!

Prove the above result, where $[X,Y]$ means the set of all homotopy classes of maps from X to Y, two topological spaces. I have answered it below.
6
votes
1answer
107 views

Tangential Space of a differentiable manifold is always $\mathbb R^n$?

Let $\mathcal M$ be a differential manifold with a point $p$. Let U be an open set, $p\in U$, on $\mathcal M$ and let $\phi,\psi:U\to \mathbb R^n$ be a charts on $\mathcal M$. I'm having diffculties ...
3
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0answers
30 views

How to tell computationally that a volume of points constitutes a manifold

Suppose that a space ${\mathbb R}^{r}$ contains a set of points which we want to consider as enclosing a volume within the space, or perhaps a volume in a submanifold (e.g., the sphere $S^{2}$ within ...
3
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1answer
135 views

Algebraic Tangent Space and Vector - an intuitive understanding?

In this question I am looking for help in understanding the Algebraic Tangent vector and what the difference is between it and the "regular" Tangent vector. A "differentiable function" near p is a ...
0
votes
1answer
90 views

Geometric Tangent Vectors - looking for and understanding of and what the point is.

The problem that I am having is that I am having quite a hard time understanding the ideas of Geometric Tangent vectors and why they are even needed - I mean one already has the usual "Tangent" of a ...
5
votes
2answers
144 views

Injectivity of a map between manifolds

I'm learning the concepts of immersions at the moment. However I'm a bit confused when they define an immersion as a function $f: X\rightarrow Y$ where $X$ and $Y$ are manifolds with dim$X <$ ...
2
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0answers
62 views

Fundamental Group, Piecewise Smooth Curves, Consevative Fields

Let M be compact Riemannian manifold. X be a vector field on M. I believe that work done by X any piecewise closed curve is zero, iff, the same is zero for a particular finite set of loops. I believe ...
3
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0answers
46 views

Maps between total spaces of holomorphic vector bundles

I am wondering what is possible and what is not possible regarding maps between the total spaces of holomorphic vector bundles. Let me outline a situation that is a bit more concrete, to help focus ...
4
votes
2answers
225 views

How do I know when a form represents an integral cohomology class?

Suppose $M$ is an $n$-dimensional manifold, and $\omega \in \Omega^p(M)$ is a closed $p$-form. Moreover, assume that $d\omega = 0$, so that it represents a de Rham cohomology class. I would like to ...
2
votes
2answers
87 views

Vector Field Generating Variation Along Curve

I'm learning a proof of the fact that length extremising curves are geodesics of the Levi-Civita connection, and have found something I don't understand. The argument states the following. Suppose ...
9
votes
1answer
144 views

Topological manifolds (dimension)

I am taking an introductory course to topology and the professor defined a topological manifold of dimension $n$ if it is hausdorff and if for every point $x$ there exists an open set $U$ around $x$ ...
3
votes
1answer
61 views

Question about diffeomorphism

Here is an assignment problem: $f:\mathbb{S}^2 \longrightarrow \mathbb{S}^2$ is smooth and surjective. Prove $\exists$ open subset $ U $ of $\mathbb{S}^2$, such that $f|_U$ is a diffeomorphism. I've ...
2
votes
1answer
53 views

Lebesgue covering dimension of a manifold

I have found many sources saying that the Lebesgue covering dimension of a (topological or smooth) manifold is the same as the dimension of the manifold. Does anyone know where I can find the proof?
4
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0answers
111 views

Differentiable manifolds, Serge Lang

I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His ...
10
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0answers
164 views

Invariant submanifolds

Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
2
votes
2answers
76 views

Smooth maps on a manifold lie group

$$ \operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\ \begin{align} &n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\ &n = 2, \operatorname{GL}_n(\mathbb ...
2
votes
1answer
81 views

question from hatcher basic 3 manifolds

The question is: why should a homologically trivial embedded sphere in a simply connected (not necessarily compact) 3 manifold M bound a compact 3 manifold embedded in M? I had this problem reading ...
3
votes
2answers
118 views

Attaching two manifolds along their boundary

I have a question about a proof in John Lee's Introduction to Topological Manifolds. Suppose $M$ and $N$ are two topological $n$-manifolds with nonempty boundary (for reference, the definition I am ...
1
vote
1answer
150 views

How to directly show that Figure 8 injective immersion is not a monomorphism

I'm working in the category of smooth manifolds. The injective immersion that takes the open unit interval $(0,1)$ to the figure 8 is a well-known example of an injective immersion that is not an ...
-1
votes
2answers
129 views

The euclidean space $\Bbb R^n$ is orientable as a manifold.

I know that The euclidean space $\Bbb R^n$ is orientable as a manifold. I think that it is orientable because it has a nowhere vanishing $n$-form. But I am not sure. Please can you explain ...
3
votes
1answer
773 views

Differential Geometry Video Lectures

I know there's a similar question here, however since what I found there wasn't what I was looking for I thought on creating a new question. I'm studying Differential Geometry through Spivak's book "A ...
2
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0answers
58 views

Stoke's theorem application to curl theorem. I did. Please can you check it?

Now, I need to apply stoke's theorem to curl theorem. My teacher gave a hint. Accourding to the hint, I accept $w=Pdy∧dz +Q dz∧dx + R dx∧dy$ $\in Ω^2(M)$ $dim(M)=2$ M is the subset of $\Bbb ...
3
votes
0answers
283 views

Real projective space is Hausdorff

I could not understand the proof of thıs proposition can you help me and give clear explanation.Just can you say how we have (n+1)x2 matrix?? This prove is correct or I need to add something ?? ...
2
votes
1answer
71 views

Real Projective Space

How I can prove this corollary 7.15.I consider that ı can apply the corollary 7.10.But I could not can you help me.
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1answer
40 views

Locally finite or not

I am tryıng to learn locally finite and can you give an explanation for my green writing please thank you
3
votes
1answer
91 views

What is overlop

I want to ask something that what is overlop.My teacher said that For Ex1, Everything is overlop hence it is not locally finite.For example 2,it doesnt overlop.Actually ı could not understand.please ...
0
votes
2answers
120 views

Topological manifold example

$\theta(x,x^2)=x$ $\Bbb X =${$(x,x^2)| x$ in $\Bbb R$} And V is subset of $\Bbb R$ $dim\Bbb X=1$ My instructor said that this is topological manifold. Why? Please can you explain me? This ...
2
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2answers
93 views

An open cover that is not locally finite

I could not understand that why is not locally finite for example 13.4 can you give me explanation please.
4
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0answers
81 views

Levi-Civita Connection for 2-dimensional Riemannian manifold

I'm trying to show the following. Suppose $(M, g)$ is a $2$-dimensional Riemannian manifold with connection $\nabla$. Suppose also that $\nabla$ is metric compatible, and that length extremizing ...
2
votes
0answers
128 views

Show that the projection map is Orientation preserving iff n is even

My question is that Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere $U =${$x∈S^n |x^{n+1} >0$}. It is a coordinate chart on ...
0
votes
1answer
72 views

Meaning of equation $dx=\sum_{A}\omega_Ae_A$.

I am reading some notes about Riemannian Structures. In definition of moving frame I see blow text and can't understand what $dx$ is: By a moving frame in $U\subseteq \mathbb{R}^N$ we mean a ...
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votes
1answer
55 views

Show that $f^{-1}(a)$ is a submanifold of $\mathbb{R}$

Let $f:\mathbb{R}\to\mathbb{R}$ be a real analytic function with infinitely many zeros. Let $a\neq 0$ be a real number. Show that $f^{-1}(a)$ is a submanifold of $\mathbb{R}$.
0
votes
1answer
106 views

Unique nearest point in epsilon neighborhood of compact real manifold?

I have to proof the following assertion: Let $X$ be a compact submanifold of $\mathbb{R}^n$ and $\mathcal{U}^\varepsilon=\{p\in\mathbb{R}^n\;:\; |p-q|<\varepsilon \text{ for some }q\in X\}$. Then ...
1
vote
1answer
124 views

I did all explanation. Can you just teach me how to calculate this interior product?

Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Show that an orientation form on $S^n$ is $w=\sum _{i=1}^{n+1}(-1)^{i-1}x^i dx^1∧...∧dx^i∧...∧dx^{n+1}$ I ...
2
votes
1answer
87 views

Manifolds with boundary and definition

Can you help for understanding this definitions in a good way.What is the my problem is that I can not image this definition and proposition in my mind and also ı dont understand the reason that ı ...
2
votes
1answer
84 views

Boundary orientation for a cylinder

Please help me.I am think that I can use stokes theorem but ı could not apply.This question is very benefical for me to learn the subject please help me :(
2
votes
1answer
65 views

Cutting out submanifolds with “orthogonal” functions

Let $Z$ be an embedded manifold in some $\mathbb{R}^M$. Then locally $Z$ is cut out by independent functions $(g_1, \ldots, g_l): \mathbb{R}^M \to \mathbb{R}^l$, where $l = \operatorname{codim} Z$. ...
0
votes
2answers
102 views

Application of the transversality theorem

I am trying to do this question in Bredon's Topology and geometry about using the transversality theorem to show that the intersection of two manifolds is a manifold. Now it goes as follows: Let ...
2
votes
0answers
45 views

Orientation-preserving diffeomorphism [duplicate]

Can you help for solving this please. Although I study this subject I could not solve this question please help me ı am willing to learn this question.
1
vote
1answer
68 views

Understanding topological and manifold boundaries on the real line

Let $M$ be the subset $[0,1)$ $∪ $ {$2$} of the real line. Find its topological boundary $\mathrm{bd}(M)$ and its manifold boundary $\partial M$. I know that to find the topological boundary, I ...
3
votes
3answers
221 views

Topological boundary vs geometric boundary

Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$ $M_2=\{(x,y) \mid x^2+y^2\le1\}$ What are the interior of $M_1$ and $M_2$ ? And What are the boundary of $M_1$ and $M_2$ ? How to find them? ...
1
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0answers
77 views

Is Cartan's magic formula applicable to time dependent vector fields?

Cartan's magic formula states: $$\mathcal{L}_v\omega = i_v\mathrm{d}\omega + \mathrm{d}i_v\omega$$ Is this also true for time dependent vector fields? If so: How can I prove it? If not: Is there a ...
4
votes
1answer
151 views

The open Möbius Band is not orientable

Can you explain my green underlying please.I have confused and ı dont understand why by the continuity of the orientation at the points $(0,0)$ and $(1,0)$ are also $e_{1},e_{2}$
4
votes
2answers
157 views

Orientations on Manifold

This is a very basic definition of orientable and very basic example 20.5 however ı could not understand definition in an good way so ı want you to explain my green writing please :) and my example ...
2
votes
1answer
142 views

Why is the cylinder surface on $\Bbb R^3$ orientable?

Why is the cylinder surface on $\Bbb R^3$ orientable? Please can someone explain me clearly?