For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
3
votes
1answer
43 views
Let $M$ and $N$ be smooth manifolds and $f: M\rightarrow N$ a diffeomorphism. Prove that the map $df:TM \rightarrow TN$ is a homeomorphism.
I am going through qualifying exam questions and I am stuck on this problem. I don't think it should be too difficult, but I am having a lot of difficulty. I am not even sure how to start. Some ...
2
votes
1answer
34 views
Why functions $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$ are called cocycles?
Let $X$ be some smooth manifold and $\{U_\alpha\}$ be its open cover. The last month I hear very often that one calls a collection of functions $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$, ...
0
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1answer
35 views
Embedded 2-Submanifold
Can you help for solving this problem ı am triying to understand embedded 2 submanifold please help me.
0
votes
1answer
93 views
Qualifying Exam Question on Manifolds
I am practicing qualifying exam problems and I am having trouble with the following question. Any help is greatly appreciated.
Let $P$ be a polygon with an even number of sides. Suppose that the ...
4
votes
2answers
93 views
Pushforward of Lie Bracket
I am trying to figure out why the following equality is true :
$$f_*[X,Y]=[f_*X,f_*Y]$$
where $f:M\rightarrow N$ is a diffeomorphism, $M$, $N$ are smooth manifolds, $X$, $Y$ are smooth vector fields ...
2
votes
1answer
43 views
Is this intuition behind product manifolds correct?
I've been studying differential geometry on Spivak's books and recently I proved that the cartesian product of manifolds is another manifold. Right, however, what's the intuition behind this? I've ...
0
votes
4answers
62 views
Why $GL(n+1,\mathbb{C})$ is compact?
I'm trying to prove that:
The set of all lines in $\mathbb{C}^{n+1}$ ($\mathbb{C}\mathbb{P}(n)$) is a complex manifold.
I'm knowing that:
If a compact group $G$ acts on $X$ transitively and ...
0
votes
2answers
81 views
Lie bracket in local coordinates. Find the formula $c^{k}$ in terms of $a^{i}$ and $b^{j}$
This is from T.U Loring's manifold book. I tried. But I didnt do the question. Please show me how to solve instructively and explicitly. I want to learn this topic. Thank you for help.
5
votes
2answers
67 views
Topological space M with partition of unity--->M paracompact. John Lee Problems
Suppose $M$ is a topological space with the property that for every open cover $X$ of $M$, there exists a partition of unity subordinate to $X$. Show that $M$ is paracompact.
1
vote
2answers
79 views
Diffeomorhism of manifold
This is one of the exam questions of the previous semester. I have studied these. But I didn't do this. Please show me how to solve this question. Thank you for help
2
votes
1answer
56 views
Tangent space at the identity element of a lie group
Let G be a lie group . we know a Lie group is a group with a smooth manifold structure s.t both the multiplication map $m$ and group inversion map $i$ are smooth .
Now by identifying ...
1
vote
1answer
42 views
Show that 2 sets are not homeomorphic
Prove that a closed interval $A=[0,1]$ and $B=\{(x,y)∈R^2 \mid ||(x,y)||≤1\}$ are not manifold
I'm struck with this problem.Can anyone explain how and what property should i use to show that for any ...
0
votes
1answer
49 views
Problem about tangent vector and the inclusion map of the unit circle.
It is so complecated for me. Please can you show how to solve. Thank you.
0
votes
0answers
45 views
How to show that open interval is manifold but closed one is not
Prove that we can define manifold's structure for
$1.$ An open interval $A=(0,1) $
$2.B=\{(x,y)\in R^2 | ||(x,y)||<1\}$
And that we can't define manifold's structure for
$3.$ An closed interval ...
8
votes
2answers
141 views
How to deal with Homeomorphisms?
I have one doubt that may be too general, I don't know, so sorry if this is not a good place to ask it. I've also seem many other people with the same problem that I have, so I think that if this ...
0
votes
1answer
54 views
The differential $i∗ : TpS_{2} → TpR_{3 }$ maps $ ∂/∂u|p,∂/∂v|p $ into $TpR_{3}. $ Find $(α_{i}, β_{i}, γ_{i})$
Hi! This was my homework. Prof. sent its answer. But I didnt understand how can this answer be reached? Please can someone explain this?
0
votes
2answers
36 views
Projective Plane and Projective Space
I have already heard of a $n$ dimensional manifold called the projective space which is the set of all lines through the origin of $\mathbb{R}^{n+1}$. Spivak presents in his Differential Geometry book ...
5
votes
2answers
155 views
Why is $\partial\partial M=\varnothing$?
Why is the border of the border of an oriented differentiable $n$-dimensional Manifold $M$ empty, that is $$\partial\partial M = \emptyset?$$
0
votes
2answers
57 views
My question is about manifold related to submersion and immersion
Let $N$ and $M$ be a manifolds of respectively dimensions $n$ and $m$. If a smooth map ( $M$ from $N$ )is an immersion at a point $p$ in $N$ then it has constant rank $n$ in a neighborhood of $p$. If ...
4
votes
2answers
107 views
geometric interpretation of Lie bracket
On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written:
We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to
which the integral ...
2
votes
0answers
55 views
Topological Manifold with ball, removed and antipodal points identified orientable?
Suppose you have a compact, orientable $(2n+1)-$manifold $M$, as in $H_{2n+1}=\mathbb{Z}$. You take a neighborhood about a point homeomorphic to $\mathbb{R}^{2n+1}$, and remove a small ball $B$. So ...
0
votes
0answers
10 views
fibered solid tori matched by a fiber preserving homeomorphism
how do I proof that two seifert fibered solid tori $V$ and $V'$ (not ordinary fibered) with the same fiber parameters matched together by a fiber preserving homeomorphism do not become a solid tori.
...
3
votes
1answer
59 views
projective cubic curve to complex projectie space
Suppose we are given the equation
$$
y^2z = x(x - z)(x - 2z)
$$
I would like to define a degree two map $g$ on this curve into complex projective space. I hate to say I am already lost here - how do I ...
0
votes
1answer
50 views
Lie bracket of vector fields on $\Bbb R^{n}$
Please show how to solve? I am stack with lie bracket. Thank you.
1
vote
1answer
58 views
show that $[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X$.
If $f$ and $g$ are $C^{∞}$ functions and $X$ and $Y$ are $C^{∞}$ vector fields on a manifold $M$, show that $[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X$
This is a proposition in a book. But I cannot prove this:(
...
0
votes
1answer
35 views
I have done the second direction of the proof. Hopefully, it is true. Please show my mistakes?
Show that two $C^{∞}$ vector fields $X$ and $Y$ on a manifold $M$ are equal if and only if for every $C^{∞}$ function $f$ on $M$,we have $Xf =Yf$.
I have sone one direction of the proof.
let $p ∈ ...
0
votes
1answer
82 views
Problem about differential of a linear map
Please can you tell how to solve this problem clearly? Please solve this explanatorily. Thank you
1
vote
1answer
42 views
Differential forms and how to show one is smooth
A class I am current taking makes some use of differential forms. In particular, we are asked to show that a certain differential form is smooth. I know in general that a differential $k$-form, ...
0
votes
1answer
69 views
About Sectional Curvature
In a paper by Yann Ollivier:
Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint
of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from ...
3
votes
2answers
107 views
Orientability of projective space
Q: Show that $\mathbb {RP}^n$ is not orientable for $n$ even.
First I looked at the definition for orientability for manifolds of higher degree than 2, because for surfaces I know the definition with ...
1
vote
1answer
62 views
How can the proof of a local theorem on a manifold involving a map with a fixed point and a differential be reduced to the case of $\mathbb{R}^n$?
Case in point: the Hartman-Grobman theorem (for maps). In the book "Geometric Theory of Dynamical Systems: An Introduction" by Palis and De Melo, the theorem is stated as follows (on page 60).
...
3
votes
0answers
82 views
Simple exercise in cohomology
I know this is a simple exercise but I am stuck unfortunately.
Question:
Use de Rham cohomology to prove that the sphere $S^2$ is not diffeomorphic to the torus $T$. You may assume that ...
2
votes
1answer
70 views
Tangent space to a product
Can you explain this question explicitly. This is a little bit difficult for me, but I want to learn how to solve. Thank you for help.
If $M$ and $N$ are manifolds, let $\pi_1:M\times N\to M$ and ...
1
vote
2answers
53 views
Compute the velocity vector.
Can you solve explicitly? please. I don't know how to solve. Thank you for help.
0
votes
1answer
47 views
Is $S$ a regular submanifold?
$M=M_{n\times n}(\Bbb R)$
$S=\operatorname{SL}(n, \Bbb R) = \left \{ A \in M \mid \det(A)=1 \right \}$
$M$ is an $n^{2}$ dimensional $C^{\infty}$ manifold.
Is $S$ a regular ...
1
vote
0answers
66 views
Determining push forward of a vector field of a submanifold
Let $M,N$ be two differentiable manifolds, $\phi:M\to N$ a diffeomorphism and $X$ a vector field on $M$. For example, one can determine the push forward
...
0
votes
1answer
41 views
Is $S$ a regular submanifold of $\Bbb R^{3}$?
$$S=\{(x,y,z) \mid x^{2}+y^{2}=z^{2}\}$$
$g: \Bbb R^{3}\to \Bbb R$, $S=g^{-1}(0)$
Is $S$ a regular submanifold of $\Bbb R^{3}$?
I'd be grateful for a clear and explicit explanation of why this is ...
4
votes
2answers
77 views
Differentiable manifolds $\mathscr C^k$ vs. $\mathscr C^\infty$
I noticed that there exists a (in some sense) better definition of the tangent space via the dual of a certain quotient algebra which is easier to work with in some cases. This however only works for ...
2
votes
0answers
37 views
Torus biholomorphic to smooth cubic curve?
I am trying to understand that all compact genus 1 Riemann surfaces are biholomorphic to a smooth cubic curve. ( assuming that $\dim H^{1,0} = \dim H^{0,1} = \frac{1}{2} \dim H^1$ )
I think I ...
1
vote
1answer
81 views
Difference between “Live” and “Define”
In many mathematical text to determine an object on manifold, the verbs "live" and "define" are used.
I'm interested to know whether there is a difference between the concepts of "to define" and "to ...
3
votes
1answer
59 views
Approach topological manifolds with smooth manifolds
Because I'm doing some problems that consider all the manifolds while the situation is really clear when considering only smooth manifolds. Thus my question is can we always appoint a topological ...
2
votes
0answers
56 views
Prove Poincare duality theorem with Morse theory.
First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical ...
1
vote
1answer
67 views
Structure on manifolds
Thank you for your attention.
First, I would like to know why we see some different structures defined on manifolds: What is the necessity to have different structures, like Kähler structure, ...
0
votes
1answer
38 views
Metric Spaces needed for Differential Geometry
I've asked here about some texts about differential geometry which doesn't assumes that the reader knows general topology. I've got good references as Do Carmo's Differential Geometry of Curves and ...
2
votes
0answers
55 views
Doubt in Spivak's examples of Manifolds
I've started to study Differential Geometry in Spivak's first volume of his Differential Geometry books. I like very much his approach since general topology isn't assumed, and since he gives many ...
1
vote
0answers
59 views
About the Morse theory
I am trying to study the Morse theory and would like to know the purpose of this study, why when we talk about the critical point of a manifold is mentioned Morse theory? , are the critical points of ...
1
vote
1answer
42 views
Doubt about the Domain of the chart on a Manifold
I have a doubt about the domain of the chart on a manifold. Suppose $M$ is a smooth manifold and that $(U, \varphi)$ is a chart on $M$, then $\varphi : U \to \mathbb{R}^n$ has $U$ as it's domain. ...
0
votes
1answer
67 views
Problem related to differential of a map
I dont understand how to solve this problem. Please can you explain the solution clearly? I want to learn how to solve such problems. Thank you
2
votes
1answer
63 views
Existence of a map homotopic to an odd degree map which is transversal to transversal submanifolds?
my question is as follows:
Given a map $f:X\rightarrow Y$ between compact spaces of equivalent dimension, and two sub-manifolds $Z_1,Z_2 \subset Y$ which transversely intersect with $I_2[Z_1,Z_2]=1$, ...
1
vote
0answers
38 views
Flow of a complex vector field?
Suppose I have a vector field X over a m-dimensional analytical manifold $M\subset \mathbb{C}^n$; how can I define the flow of $X$? Is it done in the same way as for the real case, but instead of ...
