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

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3
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1answer
313 views

Question about theorem 3.2 from Morse theory by Milnor

The demonstration of the theorem 3.2 in the book Morse theory by Milnor THEOREM $\mathbf{3.2.}$ Let $f:M\to\bf R$ be a smooth function, and let $p$ be a non-degenerate critical point with index ...
4
votes
1answer
315 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 ...
0
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1answer
102 views

What is a de Rham k-form?

I generally know what a differential k-form is. But what does it mean for a k-form to be a "de Rham" k-form? Thanks in advance!
1
vote
1answer
283 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 ...
3
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0answers
206 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 ...
9
votes
3answers
1k views

Inducing orientations on boundary manifolds

Given a $k$-manifold $M$, such that $\partial M$ is a $(k-1)$-manifold, there is a standard way in which $\partial M$ inherits the orientation of $M$. So if $M$ is oriented by the form field $\omega$, ...
2
votes
1answer
210 views

An other question about Theorem 3.1 from Morse theory by Milnor

In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that: for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla ...
3
votes
0answers
402 views

On the definition of a normal crossing divisor

I'm reading a material that states: Definition: Let F be a foliation on a analytical manifold N. A normal crossing divisor on N is a collection of submanifolds $E$ of $N$ such that for every point ...
10
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1answer
282 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
1answer
62 views

Prove $X =\left \{(x, y) \in \mathbb{R}^3 \times \mathbb{R}^3 \ | \ |x| = 1, |y| = 1, x\cdot y = \frac{1}{2}\right\}$ is a manifold

I am having trouble with the following qualifying exam problem and I would appreciate any help. Thank you. Let $X$ be the set of pairs of unit vectors $(x, y)$ in $\mathbb{R}^3$ such that $x \cdot y ...
3
votes
2answers
144 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 ...
3
votes
1answer
282 views

Given a local diffeomorphism $f: N \to M$ with $M$ orientable, then $N$ is orientable.

Given a local diffeomorphism $f: N \to M$ with $M$ orientable. Why is $N$ orientable? My professor wrote this in class without giving a proof and said "you should try to prove this for fun :)". I ...
1
vote
1answer
120 views

geometrically finite hyperbolic surface of infinite volume

I am starting to read some papers involving analysis on hyperbolic manifolds. In these the notion of a "geometrically finite hyperbolic surface of infinite volume" is mentioned frequently and I am ...
1
vote
1answer
196 views

Is this distribution involutive?

For two days I've been trying to show the following: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and consider the smooth distribution $$F=\{F_p=DR_p(e)\mathfrak{h}; p\in G\},$$ where ...
1
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1answer
96 views

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 ...
1
vote
0answers
69 views

“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 ...
2
votes
1answer
145 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 ...
1
vote
1answer
65 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
votes
1answer
136 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 ...
2
votes
1answer
67 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 $Y$ ...
1
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1answer
53 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 $$\displaystyle\nu(F)=\frac{T_pM}{F_p}.$$
4
votes
1answer
149 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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votes
1answer
1k views

Vector field on an odd sphere [closed]

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 ...
0
votes
1answer
63 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 ...
1
vote
1answer
79 views

Show the regular submanifold

Please help me how sdo I show such a problem? I Will be happy to teach me. Thank you
5
votes
1answer
333 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 ...
2
votes
1answer
224 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. ...
0
votes
1answer
82 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 ...
4
votes
2answers
71 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 ...
3
votes
1answer
169 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 ...
6
votes
1answer
279 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 ...
0
votes
1answer
84 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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vote
0answers
138 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
votes
1answer
208 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 ...
2
votes
1answer
112 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
votes
1answer
184 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 ...
3
votes
2answers
230 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
votes
1answer
77 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
159 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
53 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
190 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 ...
5
votes
2answers
703 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
72 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
80 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
94 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.
6
votes
2answers
206 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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vote
2answers
100 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
559 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
159 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
81 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.