For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
0
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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.
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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.
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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.
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2answers
52 views
Transition formula for 1-forms
I try to solve this question but ı could not.ı am working for my exam.please help me.
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 ...
1
vote
2answers
27 views
Manifold Boundary versus Topological Boundary.
Let $M$ a $n$-manifold whit boundary, i.e., for each $x\in M$, there exist $U_x\subseteq M$ open in the topology of $M$ such that $U_x$ is homeomorphic to $\mathbb{R}^n$ or homeomorphic to ...
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. ...
1
vote
0answers
22 views
Standards in P.L. Topology
About a week ago, the reading course on PL topology I'm going to follow started. The aim of the reading course is to understand the basics of PL topology and have a reasonable to good understanding of ...
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 ...
3
votes
1answer
186 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
is given in the special case whene the manifold is the Torus ,
My question is : can i prove it in the case where the ...
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 ...
0
votes
1answer
26 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
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 ...
5
votes
2answers
96 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
127 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 ...
1
vote
0answers
43 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 ...
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
1answer
42 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 ...
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 ...
3
votes
1answer
42 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
39 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
37 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
vote
1answer
58 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
34 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 ...
1
vote
1answer
48 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
25 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 ...
1
vote
0answers
18 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
46 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
vote
1answer
33 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
55 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 ...
0
votes
1answer
419 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
...
0
votes
1answer
27 views
Lie bracket in local coordinates
Can you help for solving this.I have an manıfold exam and ı am working but ı have a problem about lie bracket.
And ı am putting what ı did..
1
vote
1answer
49 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
127 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
59 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
44 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
48 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
66 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 ...
5
votes
1answer
132 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
44 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 ...
1
vote
0answers
54 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
118 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
50 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
76 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
87 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 ...
