Tagged Questions
1
vote
2answers
36 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 ...
3
votes
1answer
45 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 ...
4
votes
0answers
53 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
votes
0answers
70 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
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 ...
1
vote
1answer
23 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
99 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
64 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
votes
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
vote
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.
1
vote
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
73 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
65 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 ...
1
vote
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
votes
0answers
40 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
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 ...
1
vote
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
votes
1answer
53 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
48 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
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.
1
vote
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 ...
-1
votes
0answers
35 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? ...
4
votes
1answer
71 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
84 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
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
votes
1answer
73 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
58 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 - ...
2
votes
1answer
59 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 ...
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 ...
0
votes
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 ...
2
votes
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.
2
votes
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).$$
I am confused so much. I ...
0
votes
1answer
49 views
Partitions of unity and bump function
I can not image this guestion in my mind.can you give me graph and help how ı can prove this question please.
0
votes
1answer
41 views
How do I show that $F^{∗}(dx∧dy∧dz) = ρ^{2} \sin φ dρ∧dφ∧dθ$.
I dont know how to solve. Please help me. I need to understand such types of the question for my exam studyings.
0
votes
1answer
60 views
Find a $1$-form $ω$ on $\mathbb R^2 −\{(0,0)\}$ such that $ω(X) = 1$ and $ω(Y) = 0$.
Please ı dont know what I need to do. thus, help me to solve.
2
votes
1answer
79 views
When is a topological space a manifold?
I'm looking for someone to point me in the direction of papers or books which discuss when a topological space (perhaps with the conditions locally compact, Hausdorff) is a topological manifold, ...
0
votes
1answer
33 views
prove that $supp(π^{∗} f) = (supp f)×N.$ Please can you check my answer? Also more explanation please.
My question is that
Let $f \colon M \to R$ be a $C^{\infty}$ function on a manifold $M$.
If $N$ is another manifold and $π \colon M \times N \to M$ is the projection onto the first factor, prove ...
1
vote
0answers
42 views
Deformation retract
How to prove that $r_t$ is a deformation retract
$M^a=\lbrace q\in M ; f(q)\leq a\rbrace$
We have the definition :
$r_t$ is a difformation retract if:
$r_t$ is a continius ,onto application ...
1
vote
0answers
23 views
Equivalence class involving Lie Brackets..
Can anyone help me with a proof, I spent hours on it and nothing: I want to show $$\overline{[fX, Y]}=f\overline{[X, Y]},$$ where $X$ and $Y$ are smooth vector fields on a smooth manifold $M$, $f\in ...
2
votes
1answer
56 views
Incomplete vector field
Is there a way I can tell if a vector field on a manifold or just $\mathbb{R}^n$ is incomplete simply by just looking at its formula. For instance on $\mathbb{R}$, $\displaystyle X= (x^2+1) ...
2
votes
1answer
42 views
Many partitions of unity on sufficiently “nice”; what does this mean?
In a class I am taking, we are told that are manifolds have "many" partitions of unity (we assume paracompact, Hausdorff, second-countable). However, the course content is not related to this subject. ...
2
votes
1answer
67 views
“Completing” a vector field on a non-compact manifold $M$
Suppose I have a non-compact smooth manifold $M$ and an arbitrary nowhere-vanishing smooth vector field $X$ on $M$ which is not complete.
Is there a way to create a smooth vector field $V$ that is ...
4
votes
1answer
58 views
Differential of smooth function on manifold
In the book I am using, the author defines differentials in the following way.
Given smooth manifolds $M,N$ and a smooth mapping $\psi:M\to N$ define the differential $d\psi_m$ at a point $m\in M$ as ...
1
vote
1answer
89 views
Complete non-vanishing vector field
Let $M$ be a non-compact smooth manifold. Suppose we have a nowhere-vanishing smooth vector field X. Is this vector field complete?
I know it is when $M$ is compact. However, I am unsure in the ...
2
votes
0answers
33 views
Submanifold with boundary of a manifold with boundary
Let $M$ be a smooth manifold.
(1) A subset $S$ of $M$ that with the subspace topology is a topological manifold (with or without boundary), together with a differential structure that makes the ...
8
votes
0answers
165 views
Infinite dimensional constant rank theorem
Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set ...
2
votes
2answers
52 views
Extending a smooth map
When can I extend a smooth map $f:\mathbb{R^2}-\lbrace 0 \rbrace \to S^1$ to a smooth map $\tilde{f}:\mathbb{R^2} \to S^1$. For instance, consider $g(x,y)=(x,y)/\sqrt{x^2+y^2}$? Am I able to extend ...
