For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
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1answer
10 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!
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1answer
25 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 ...
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0answers
20 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 ...
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0answers
10 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 $\omega$, ...
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1answer
34 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 ...
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0answers
27 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 ...
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0answers
42 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 ...
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1answer
38 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 ...
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votes
2answers
31 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
105 views
Diffeomorphism without fixed points [duplicate]
Suppose I have a nowhere-vanishing vector field $X$ on a smooth compact manifold $M$. I am trying to prove that there is some diffeomorphism $f:M \to M$ such that for all $p \in M$, $f(p) \neq p$.
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3
votes
1answer
29 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 ...
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1answer
34 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
31 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
55 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 ...
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0answers
33 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 ...
