1
vote
0answers
36 views

Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$ \phi: M \to N. $$ As far as I understand, this gives rise to two distinct isomorphisms $$ a : \mathcal A^1(M) \cong \mathcal A^1(N) :b $$ between the space of ...
7
votes
2answers
169 views

Interpretation of $p$-forms

Let $M$ be a smooth manifold, let $C^{\infty}(M)$ be set of all smooth functions from $M$ to $\mathbb R$ and let $Vec(M)$ denote the set of all vector fields on $M$. A $1$-form on $M$ is a ...
4
votes
0answers
74 views

Generalizing the Hopf invariant to arbitrary manifolds

I recently ran across a qual question about a generalization the Hopf invariant to smooth maps $f: M^{4n-1} \to N^{2n}$ between arbitrary closed connected oriented manifolds of the indicated ...
5
votes
1answer
105 views

Poincare dual of unit circle

I'm trying to self-study Differential Forms in Algebraic Topology by Bott and Tu. I've come across this exercise: Show that the closed Poincare dual of the unit circle in $ R^2-\{0 \} $ is zero, ...
1
vote
1answer
89 views

Change of Coordinate Formula for Differential Forms

Let $M$ be a manifold, $x$ local coordinates on an open set $U$, $y$ local coordinates on an open set $V$. In addition, let $(x, \alpha)$ and $(y, \beta)$ be two induced bases for the common part of ...
4
votes
0answers
62 views

A question on harmonic two-forms

Let $(M^4,g)$ be a closed Riemannian four-manifold with $b_2^+>0$ and $b_2^->0$, is it possible to find two harmonic two-forms $\alpha\in H^2_+(M)$ and $\beta\in H^2_-(M)$, such that ...
0
votes
1answer
43 views

When is an exact 2-form harmonic?

Let $\alpha$ be an exact two-form, $\alpha=d\beta$ for some one-form $\beta$, when is $\alpha$ harmonic? By uniqueness of harmonic forms in cohomology classes, it cannot be harmonic?
2
votes
1answer
102 views

De Rham cohomology, and forms on manifolds

In String Theory and M-Theory by Becker, Becker and Schwarz, they introduce a group, $$C^{p}(M)$$ which they denote the group of all closed $p$-forms on the manifold $M$. Furthermore, they state ...
4
votes
0answers
61 views

show that $\omega$ is exact if and only if the integral of $\omega$ over every p-cycle is 0

Let $M$ be an oriented smooth manifold and $\omega$ a closed $p$-form on $M$. Show that $\omega$ is exact if and only if the integral of $\omega$ over every $p$-cycle is $0$. In particular, how to ...
0
votes
0answers
54 views

Proving $d(f^*e_i')=0$

Let $f:M\to N$ be a differentiable map between two manifolds. $e_i$ is a basis vector of $N$ with respect to some chart and $e_i'$ its dual (i.e. $e_i'(e_j)=\delta_{ij}$). How do I prove the ...
6
votes
2answers
156 views

Why 'closed differential forms' are called 'closed'?

As is well known a differential form $ \omega $ is called closed differential form if it satisfies $ \mbox{d} \omega = 0 \, \, (\ast) \, $ where $ \mbox {d} $ is the exterior derivative. I think the ...
2
votes
0answers
58 views

The first proof for Poincare lemma in history

How can I get a reference about the first proof of Poincare lemma in history? I already know some methods of proof, but I do want to know the original approach. Thanks for your help!
1
vote
1answer
48 views

Proving Borsuk-Ulam with Stokes

What is the easiest way to deduce the Borsuk-Ulam theorem in the case $n=2$ by using integration on manifolds and Stokes theorem? So I want to prove the following: Given a map $f\colon S^2\rightarrow ...
0
votes
1answer
77 views

Condition for Differential Forms to Pass to the Quotient

everyone: I was reading this question : What do we mean when we say a differential form "descends to the quotient"? which is related to mine. But the reply given did not answer my question ...
0
votes
0answers
63 views

Is a Differential 1-form ( on M^3)Dual to a Contact Vector Field a Symplectic Form?

say $w$ is a global contact 1-form on a 3-manifold $M^3$ , meaning $w \wedge dw \neq 0$ at any point in the manifold , and let $X$ be the vector field dual ( under, say, a choice of Riemannian metric ...
2
votes
1answer
129 views

About Stokes' theorem

I noticed that in the proof of Stokes' theorem on manifolds, the condition that the form $\omega$ is compactly supported ensures that the sum is finite so that we can change the order of sum and ...
0
votes
1answer
127 views

Finding Reeb Vector Field Associated with a Contact Form

I would greatly appreciate it if you could help me with the following: I'm curious as to how to find the Reeb field $R_w$ associated to a specific contact form $w$; does one actually find $R_w$ as ...
5
votes
1answer
166 views

1- forms on a torus

I think this is a very simple question but I'm not really confident in mathematics (even if I like it very much) Let's fix a cube $[0,1]^3$ in $R^3$ and identify opposite sides, so as to construct a ...
3
votes
3answers
102 views

Inner product of De Rham cohomology classes

Is there a well-defined inner product between cohomology classes? In particular, is it possible to extend the Hodge inner product? If I try, I obtain this: $$\int *(\omega + d\lambda)\wedge (\sigma + ...
3
votes
1answer
108 views

What do we mean when we say a differential form “descends to the quotient”?

Let $S$ be a surface and let $f:S\rightarrow S$ be a diffeomorphism. We define the mapping torus $M_f$ of the pair $(S,f)$ to be the quotient $$(S\times I) /\sim \quad \text{ where } \ (1,x) \sim ...
0
votes
1answer
28 views

Question on 2-chain on $\mathbb{R}^3$

Let $\gamma:[0,1]\to\mathbb{R}^3\setminus\{0\}$ be a simplex, with $\gamma(0)=\gamma(1)$. How can I show that exists a $2$-chain $\sigma$ on $\mathbb{R}^3\setminus\{0\}$ such that ...
2
votes
2answers
69 views

Why is $\theta \not \in C^{\infty}(S^1)$?

Why is $\theta \not \in C^{\infty}(S^1)$? I know that since $\int_{S^1} d\theta = 2\pi$ then $d\theta$ is not exact. Thus since $d(\theta)=d\theta$, $\theta$ must not be $C^{\infty}$ but it seems ...
3
votes
0answers
98 views

Is this a trivial Stokes exercise?

I'm working on this exercise from an old Spivak Differential Geometry book which I must be misunderstanding. The question reads: Let $M$ and $N$ be compact $n$-dimensional manifolds with boundary and ...
0
votes
1answer
52 views

Restricting the domain of an integral on a manifold

I would like to prove the following: Guess. Suppose M is an orientable smooth manifold without boundary and W is an open set in M. If $\omega$ is a smooth n-form on M such that $\operatorname{supp} ...
4
votes
1answer
228 views

Are these two definitions of exterior derivative equivalent?

I saw two definition of the exterior derivative of a $k$-form $\omega$. First definition: $$(d\omega)_p(v_0,\ldots,v_k)= \sum_{i=0}^k(-1)^id(\omega(v_0,\ldots,\hat{v_i},\ldots,v_k))_p(v_i)$$ Second ...
5
votes
1answer
136 views

Diffeomorphic riemannian manifolds and volume forms

Maybe the question will be stupid, but I'm a beginner in riemannian geometry... We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
8
votes
1answer
408 views

Closed not exact form on $\mathbb{R}^n\setminus\{0\}$

I'd like to construct a closed but not exact $n-1$-form $\omega$ on $\mathbb{R}^n\setminus\{0\}$ in analogy to the winding form: $$\frac{x~dy-y~dx}{x^2+y^2}$$ I think something like ...
4
votes
1answer
129 views

Highest DeRahm Cohomology

Let $X$ be a $C^\infty$ manifold, compact oriented and connected of dimension $n$. How do you prove that the integration map $$\int_X: \omega \mapsto \int_X \omega $$ from $H^n_{DR}(X)$ to ...
5
votes
1answer
215 views

Pullback of differential form on the double covering

On a double covering there is a differential form $\omega$ arises by the pullback of a differential form under the projection iff it is the pullback of $\omega$ under the map $i$, where $i$ is the map ...
0
votes
1answer
148 views

Averaging differential forms

Let $M$ be a manifold with a circle action, i.e. with a 1-parameter group of diffeomorphisms $\phi_t:M\to M$ of period 1. I think of the average of a differential form $\omega \in \Omega^n(M)$ with ...
6
votes
2answers
493 views

Poincaré Lemma Contractible Hypothesis

Poincaré's Lemma is often stated as saying that a closed differential form on a star-shaped domain is exact. More generally, it is true that a closed differential form on a contractible domain is ...
3
votes
2answers
105 views

Proving that a particular submanifold of the cotangent space is Lagrangian

I have the following problem in my differential topology class: Let $M$ be an $n$-dimensional manifold and let $\omega$ denote the standard symplectic form on the cotangent space $T^*M$. Let $f \in ...
5
votes
1answer
279 views

$4$-form $ \omega \wedge \omega$ vanishes on $S^4$

If $\omega$ is a closed $2$-form on $S^4$, how can I show the $4$-form $ \omega \wedge \omega$ vanishes somewhere on $S^4$? I am guessing that the fact we're talking about the $2$-form being ...
2
votes
1answer
214 views

Differential forms on a $S^1$-manifold

I am reading about differential forms on manifolds with group actions and there is an 'obvious' formula which I don't quite understand. Let $X$ be a manifold with a smooth circle action, that is a ...
4
votes
1answer
576 views

Bundle orientability vs manifold orientability

Given a vector bundle, I am a bit hazy about the difference between the notions of its orientability as a bundle and as a manifold. I think I know that the following are true, A tangent bundle of a ...
1
vote
1answer
206 views

If $S^{2n+1}$ is covering space of $X$, then $X$ is orientable.

Is there any direct way to prove that $n$-manifold is orientable? In AT we can just calculate $n$'th homology group and check whether it's $\mathbb Z$ or $0$. But I want a geometric method, using ...
4
votes
1answer
233 views

Exercise concerning the Lefschetz fixed point number

I can't see a good approach to the third part of the following problem: Let $f: M \to M$ be a smooth map of a compact oriented manifold into itself. Denote by $H^q(f)$ the induced map on the ...
8
votes
2answers
426 views

Examples of Computations in Algebraic Topology

I have started reading "Differential Forms in Algebraic Topology" by Bott, Tu, recently. While I'm quite happy with the exposition of the theorems and explanation of theoretical results, I'm missing ...