For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
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2answers
31 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 ...
8
votes
1answer
71 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
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1answer
44 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 ...
1
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0answers
17 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
48 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 ...
6
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0answers
66 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
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2answers
50 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
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1answer
52 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
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2answers
73 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
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1answer
22 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
98 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 ...
2
votes
1answer
63 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
33 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 ...
2
votes
0answers
51 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 ?? ...
1
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1answer
53 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
31 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
72 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 ...
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0answers
47 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 ...
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2answers
54 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
39 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
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0answers
74 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
49 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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1answer
45 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}$.
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1answer
43 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
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1answer
108 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
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1answer
52 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
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1answer
47 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
33 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$. ...
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votes
2answers
42 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
36 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.
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0answers
66 views
explanation of examples related to Boundary Orientation.
I found the example from my textbook. I understood similar example related to boundary orientation on $∂ H^n$ But I could not understand these two example which I posted. Please can you explain me ...
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1answer
33 views
Now I am asking that the topological and manifold boudary for real line I am grateful to explain me more clearly and instructively.
Let M be the subset $[0,1[$ $∪ $ {$2$} of the real line. Find its topological boundary $bd(M)$ and its manifold boundary $∂ M$.
I know that while I find the topological boundary, I need to show ...
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0answers
34 views
on flows of two vector fields
this is the last part of a homework question. I got some problem understanding the question itself, wondering if anyone can help me with this part.
On manifold $R^2-\{0\}$, define two vector fields ...
3
votes
3answers
133 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? ...
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0answers
30 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
69 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
83 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
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1answer
81 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?
0
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1answer
71 views
Orientation preserving diffeomorphism.
I am stuck with the question. I guess that I need to write jacobian matrix. But I could not do. Please help me thank you
4
votes
2answers
57 views
Proving homeomorphism between surface and $\mathbb{R}^2$ minus Cantor Set
I've been working with Spivak's Differential Geometry exercises and I found myself confused with this one: "Let $C\subset \mathbb{R} \subset \mathbb{R}^2$ be the Cantor set. Show that $\mathbb{R}^2 - ...
1
vote
1answer
48 views
Problems on submanifolds
I am learning differential geometry and a basis of the theory of smooth manifolds but i'm feeling a lack of practice in solving problems on submanifolds in $\mathbb{R}^n$ (problems like 'prove that ...
2
votes
1answer
58 views
Why is the diffential of a map between manifolds a map between the tangent spaces?
In the books that I have seen, given a smooth map $\phi: M \rightarrow N$ where $N$ and $M$ are manifolds, the differential at a point $x$ is defined as $d \phi_x: T_x M \rightarrow T_x N$. Why is it ...
0
votes
1answer
41 views
Function from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$ - Higher dimensions
If we have a function such as: $\mathbb{R}$ to $\mathbb{R}$, $f(x)=x$, we have a one varible function, that is living in a two dimensional space, because n+n = 1+1 = 2, so, if we have the following ...
1
vote
1answer
62 views
Show that the zero set of $f$ is an orientable submanifold of $\Bbb R^{n+1}$.
Suppose $f(x_1,...,x_{n+1})$ is a$ C^∞$ function on $\Bbb R^{n+1}$ with $0$ as a regular value. Show that the zero set of $f$ is an orientable submanifold of $\Bbb R_{n+1}$. In particular, the unit ...
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1answer
33 views
How to decide whether F is orientation-preserving or orientation-reversing as a diffeomorphism onto its image.
Let $U$ be the open set $(0,∞)×(0,2π)$ in the $(r,θ)$ -plane $R^2$.
We define $F : U ⊂ R^2 → R^2$ by $F (r, θ ) = (r cos θ , r sin θ )$.
How to decide whether F is orientation-preserving or ...
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0answers
71 views
Figure $\infty$ is immersion of circle
Where can I find prove of:
Figure $\infty$ is immersion of circle.
More thanks for a prove or a function between these manifolds.
15
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0answers
135 views
If $S\times\mathbb{R}$ is homeomorphic to $T\times\mathbb{R}$, and $S$ and $T$ are compact, can we conclude that $S$ and $T$ are homeomorphic?
If $S \times \mathbb{R}$ is homeomorphic to $T \times \mathbb{R}$ and $S$ and $T$ are compact, connected manifolds (according to an earlier question if one of them is compact the other one needs to be ...
3
votes
1answer
51 views
Classifying Vector Bundles
Given a manifold $M$, is there a way of classifying up to isomorphism all possible vector bundles over $M$ of a given rank? Some other questions on this site deal with specific cases, which all seem ...
3
votes
2answers
48 views
Algebraic surface as a smooth manifold
Let $S$ be the set of points $x=(x_1,x_2,\ldots,x_9)\in \mathbb{R}^9$ which satisfy the following conditions:
$$x_1^{2}+x_2^{2}+x_3^{2}=x_4^{2}+x_5^{2}+x_6^{2}=x_7^{2}+x_8^{2}+x_9^{2}=1$$
...
2
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1answer
56 views
$F(x,y) = (x^2 +y^2,xy)$. compute $F^{∗}(u \, du+v \, dv)$
Let $F : \Bbb R^2 → \Bbb R^2$ be given by
If $u$,$v$ are the standard coordinates on the target $\Bbb R^2$,
compute $F^{∗}(u \, du+v \, dv)$.
$$F(x,y) = (x^2 +y^2,xy).$$
I am confused so much. I ...
