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

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Question about lie bracket..

Let $G$ be a Lie group with Lie algebras $\mathfrak{g}$ and let $\mathfrak{h}\subseteq \mathfrak{g}$ be a Lie subalgebra. Write $F_p=DR_p(e)\mathfrak{h}$, $p\in G$, where $R_p:G\rightarrow G$ given by ...
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0answers
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“Rational grids” on manifolds.

Here is something which is bothering me a bit. You have rationals on a line. You can define a rational grid on R^n by taking points with all coordinates having rational values. Is there a ...
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1answer
135 views

Poisson bracket of coordinates

I just derived that in local coordinates (it suffices to centre) around $0$, that $$\{f,g\}(x)=\sum_{i,j}\{x^i,x^j\}\frac{\partial f}{\partial x^i}\frac{\partial g}{\partial x^j}$$ only using the ...
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1answer
57 views

Can you express nice manifolds as the zero locus of functions with linearly independent derivatives?

Let ${\bf f}:U\to \mathbb R^{n-k}$ be a continuously differentiable function. Then ${\bf f}^{-1}(0)$ is a manifold if $[{\bf D}{\bf f}(x)]$ is surjective at all $x$. This is equivalent to the ...
2
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1answer
108 views

Bott connection

Can anyone help me showing the following: Let $E$ be a smooth vector bundle over $M$ and $F\subseteq TM$ a smooth distribution. A $F$-connection is a $\mathbb R$-bilinear aplication ...
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1answer
66 views

Given that $X$ is closed and $Y$ is connected, prove that $Y$ is also closed.

I am having trouble with the following qualifying exam problem. Suppose $f: X \rightarrow Y$ is a smooth immersion between smooth manifolds of the same dimension. Given that $X$ is closed and $Y$ ...
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1answer
49 views

Distribution and Tangent Bundle

Let $F=\{F_p; p\in M\}\subseteq TM$ be a rank $k$ smooth distribution. Can anyone explain-me what is the set $$\displaystyle\nu(F)=\frac{T_pM}{F_p}.$$
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1answer
138 views

Vector Bundle Doubt..

Well I have a doubt about a rank $k$ vector bundle. My definition of vector bundle is: A rank $k$ vector bundle is a triple $(\pi, E, M)$ where $E$ and $M$ are smooth manifolds and $\pi:E\rightarrow ...
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1k views

Vector field on an odd sphere

Let $x^1,y^1,\ldots,x^n,y^n$ be the standard coordinates on $\mathbb{R}^{2n}$. The unit sphere $S^{2n-1}$ in $\mathbb{R}^{2n}$ is defined by the equation $\sum_{i=1}^n(x^i)^2+(y^i)^2=1$. Show that ...
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1answer
56 views

Lie bracket in local coordinates.

$\bf 14.9.$ Lie bracket in local coordinates Consider the two vector fields $X,Y$ on $\mathbb{R}^n$: $$X=\sum a^i\dfrac\partial{\partial x^i},\qquad Y=\sum b^j\dfrac\partial{\partial x^j},$$ where ...
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1answer
74 views

Show the regular submanifold

Please help me how sdo I show such a problem? I Will be happy to teach me. Thank you
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1answer
295 views

Constant Rank theorem for domain with nonempty boundary

Problem 4-3 in J.M. Lee's introductory text about smooth manifolds, asks to formulate and prove a version of the constant rank theorem for a map of constant rank whose domain is a smooth manifold with ...
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1answer
189 views

Orthogonal group is a regular submanifold of $GL(n,\Bbb R)$

I want to show that $O(n)$ is a regular submanifold of $GL(n,\Bbb R)$. I think that I can use constant rank theorem but how? I am putting the picture that what I did. Please help me I want to learn. ...
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1answer
73 views

Finding the kernel of Pushforward of $f:\mathbb R^n\rightarrow \mathbb R^k$

Let $U$ be an open subset of $\mathbb R^n$, $f:U\rightarrow\mathbb R^k$ a smooth map such that its pushforward is onto, for each $x\in U$, i.e. $$f_{*x}:T_xU\rightarrow T_{f(x)}\mathbb R^k$$ is ...
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2answers
66 views

The zero set of $z_0^2+z_1^2-1$ in $\mathbb{C}^2$.

Recently, I read the notes "Vector bundles on Riemann surfaces" by Sabin Cautis (http://www-bcf.usc.edu/~cautis/classes/notes-bundles.pdf). On the sixth page of these notes, there is a statement ...
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1answer
141 views

Smooth Structure of the Torus

Consider the torus $T^2=S^1\times S^1$(where $S^1$ is the unit circle centered at $0$ in $\mathbb C$). Define a smooth structure on $S^1$ and $T^2$. ($\checkmark$) Let $f:T^2 ...
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1answer
245 views

Uniqueness of Smoothed Corners

Let $M^m$ be a smooth manifold with boundary. We may form a new topological manifold by "adjoining a handle to $M$", I.e. by choosing a smooth map $h : \partial \mathbb{D}^{\mu} \times ...
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1answer
79 views

Definition of a submanifold

Say what it means for a system of equations $$f_1 = \cdots = f_m = 0,$$ where $f_i(x_1, \cdots x_n)$ are differentiable functions, to define a submanifold near a point $a = (a_1, \cdots ...
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0answers
126 views

Sobolev trace theorem for manifolds with boundary

Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality $$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$ will hold. ...
3
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1answer
188 views

Lemme 2.4 in Morse theory by Milnor

This is lemma 2.4 from "Morse theory" by Milnor ,with the prove I have some questions about this prove : 1) why $\displaystyle\frac{dc}{dt}(f)=\lim_{h\rightarrow 0} \frac{fc(t+h)-fc(t)}{h}$ and ...
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1answer
95 views

Doubt about $n$-holed Torus and Handles

I have a doubt on the construction of the $n$-holed torus as seen on Spivak's Differential Geometry book. Spivak gives a very good argument on how to construct it: take the usual torus ...
2
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1answer
155 views

Gradient of a functional

Given a compact manifold with a Riemannian metric $g$, we define the total scalar curvature by $$E(g)=\int_M RdV$$ Let us consider the first variation of $E$ under an arbitrary change of metric. We ...
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2answers
192 views

About the definition of tangent space of smooth manifold

For a smooth manifold $\mathscr M$ I have seen following definition for the tangent space at a point $m\in\mathscr M$. Define it to be $(F_m/F_m^2)^*$, where $F_m$ denotes the set of germs of smooth ...
3
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1answer
69 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 ...
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1answer
112 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 $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$, ...
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1answer
49 views

Embedded 2-Submanifold

$\bf 1.7$ Submersions. Quotient Manifolds Problem $\bf1.7.1\;$ Let $f:\mathbb R^3\to\mathbb R$ be given by $f(x,y,z)=x^2+y^2-1.$ $\quad(1)$ Prove that $C=f ^{-1}(0)$ is an embedded ...
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1answer
181 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 ...
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2answers
474 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 ...
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1answer
64 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 ...
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4answers
78 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 ...
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2answers
93 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.
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2answers
177 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.
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2answers
98 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
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1answer
468 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 ...
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1answer
135 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 ...
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1answer
75 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.
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1answer
215 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 ...
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2answers
762 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 ...
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1answer
73 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?
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2answers
197 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 ...
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2answers
184 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 $$\partial\partial M = \emptyset?$$
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198 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 ...
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2answers
358 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 ...
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1answer
93 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 ...
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1answer
81 views

Lie bracket of vector fields on $\Bbb R^{n}$

Please show how to solve? I am stack with lie bracket. Thank you.
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1answer
245 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:( ...
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1answer
42 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 ∈ ...
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1answer
135 views

Problem about differential of a linear map

Please can you tell how to solve this problem clearly? Please solve this explanatorily. Thank you
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1answer
98 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, ...
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1answer
208 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 ...