For questions about differential forms which commonly arise in differential geometry, and sometimes in multivariable calculus.

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0
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2answers
105 views

Writing a diffEQ as $P(x,y)dx + Q(x,y)dy = 0$ instead of in terms of $dy/dx$

I'm reading in Tenenbaum and Pollard's Ordinary Differential Equations where they introduce the concept of the differential. Suppose $y=f(x)$ is differentiable. He defines the differential by $dy(x, ...
4
votes
0answers
81 views

Criterion of nonsingular varieties

It's well-known fact, that if $X$ is non-singular algebraic variety over algebraically closed field $k$ and $Y \subset X$ is its irreducible closed subscheme defined by sheaf of ideals $J$, then $Y$ ...
6
votes
2answers
167 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 ...
0
votes
1answer
62 views

$V, W$ vector fields, $\omega$ one-form, $d\omega(V, W)=?$

$V, W$ vector fields, $\omega$ one-form, what is the definition of $d\omega(V, W)$? I have seen some equalities and I suppose that $d\omega(V,W)=(V^jW^i-V^iW^j) \frac{\partial f_i}{\partial x^j}$ ...
10
votes
5answers
282 views

Is line element mathematically rigorous?

I know differentials (in a way of standard analysis) are not very rigorous in mathematics, there are a lot of amazing answers here on the topic. But what about line element? $$ds^2 = dx^2 + dy^2 ...
2
votes
0answers
66 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

If differential 1-forms agree on chains with integer coefficients, are they equal?

Let $M$ be a real, smooth manifold. Let $\omega_1$ and $\omega_2$ be differential 1-forms on $M$, and let $C_1(\mathbb Z,M)$ denote the set of 1-chains with integer coefficients. If \begin{align} ...
1
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0answers
27 views

Litterature: Convex Geometry using differential forms

I've been looking for papers or books that discuss/use differential forms in convex geometry. I have been very unsuccessful in my search and was wondering if anybody here had come across such ...
8
votes
1answer
227 views

When can a functional be written as the integral of a 1-form?

Let a real, smooth manifold $M$ be given. Let $\Gamma$ denote the set of all path segments on $M$, namely the set of all paths of the form $\gamma:[a,b]\to M$. Let $Q:\Gamma\to\mathbb R$ be a ...
5
votes
1answer
128 views

Pullback of a form using the Hopf fibration

I embed the 2-sphere and the 3-sphere in $\mathbb{R}^3$ and $\mathbb{R}^4$ respectively. Then denote by $\{x_1,x_2,x_3,x_4\}$ the coordinates on the 3-sphere and $\{y_1,y_2,y_3\}$ on the 2-sphere. So ...
2
votes
1answer
130 views

Exercice on a differential form

Let $\omega$ be a $q$-form on $\mathbb{R}^2$ and let $Z_{\mathbb{R}^2}(dx_1)=\{p \in \mathbb{R}^2 \colon (dx_1)_{|p}=0\}$ $Z_{S^1}(dx_1)=\{p \in S^1 \colon (dx1_{|S^1})_{|p}=0\}$ where ...
3
votes
2answers
152 views

Explanation for the integral of differential forms

In our course of differential geometry we defined the integral $\int_{U} \omega$ of a differential form $\omega=f dx_1\wedge \ldots \wedge dx_n: T^nU \rightarrow \mathbb R$ with $U\subseteq \mathbb ...
1
vote
0answers
67 views

Structure equations on the 3-sphere

On the 3-sphere I have found the vector fields $X_1=(-x_2,x_1,-x_4,x_3)$, $X_2=(-x_3,x_4,x_1,-x_2)$, $X_3=(-x_4,-x_3,x_2,x_1)$, in the basis $\left\{\frac{\partial}{\partial ...
3
votes
2answers
83 views

What is the value of the 2-form $X(u(Y))-Y(u(X))-u([X,Y])$, is it $2du(X,Y)$?

Let $X,Y$ be two vector fields on a Riemannian manifold $(M,g,\nabla)$ where $\nabla $ is the symmetric metric connection on $M$ and let $u$ be a 1-form. I want to find the value or a simple form or ...
2
votes
1answer
55 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
90 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 ...
1
vote
3answers
125 views

Prove $d(f\alpha)=d(f \wedge \alpha)$

I am reading the article http://en.wikipedia.org/wiki/Exterior_derivative and a definition of an exterior derivative from Axioms for the exterior derivative. How could I show that if $f$ is a function ...
5
votes
1answer
108 views

How do Chern classes behave under connected sums?

I often see people talking about Chern classes of manifolds like $\mathbb{CP}^2 \# \mathbb{CP}^2$, or $\mathbb{CP}^2\#\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$, where $\#$ denotes connected sum and the ...
1
vote
1answer
129 views

What is the one form given its value for a vector field?

I read an article on vector fields. the author defined a 1-form on a manifold $M$ as $u(X)=\rho$ when $X$ is a given vector field and $\rho$ is a given real valued function defined on $M$. can we ...
1
vote
3answers
68 views

Differential closed form

Im trying to go alone through Fultons, Introduction to algebraic topology. He asks whether there is a function $g$ on a region such that $dg$ is the form: $$\omega =\dfrac{-ydx+xdy}{x^2+y^2}$$ in some ...
2
votes
2answers
172 views

Integration of a 2-form

$\textit{What is}$ $\int_C{\omega}$ $\textit{where}$ $\omega=\frac{dx \wedge dy}{x^2+y^2}$ $\textit{and}$ $C(t_1,t_2)=(t_1+1)(\cos(2\pi t_2),\sin(2\pi t_2)) : I_2 \rightarrow \mathbb{R}^2 - ...
0
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0answers
68 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 ...
1
vote
2answers
119 views

Question about differential form

$\omega = y dx + dz$ is a differential form in $\mathbb{R}^3$, then what is ${\rm ker}(\omega)$? Is ${\rm ker}(\omega)$ integrable? Can you teach me about this question in details? Many thanks!
2
votes
2answers
70 views

If $\omega$ is closed on $\mathbb R ^2 - 0$ and $\text d \omega =0$, then $\omega = \text d g+ \lambda \text d \theta$.

I'm trying to solve problem 4-30 from “Calculus on manifolds”, which is the one in the title, where $$\text d \theta = -\dfrac{y}{x^2+y^2}\text d x+\dfrac{x}{x^2+y^2}\text d y.$$ I think I'm on the ...
1
vote
1answer
91 views

$df$ vanish in a compact manifold in at least 2 points

I need to prove that if $M$ is a compact manifold and $f$ is a smooth function in $M$, then $df$ vanish in at least 2 different points of $M$. I don't know where to start. Any suggestion will be ...
4
votes
0answers
176 views

Top de Rham cohomology

I just realized that I never really understood why $H_{dR}^n(M, \mathbb{R}) = \mathbb{R}$ if $M$ is compact and $H_{dR}^n(M, \mathbb{R}) = \{0\}$ if $M$ is not compact (provided that's true?). I'm ...
2
votes
1answer
67 views

Product integration of differential forms

Let $\alpha, \beta$ two forms continuous with compact supports and maximum degree on surfaces oriented $M, N $ respectively. consider $\pi_M:M\times N \rightarrow M$ and $\pi_N:M\times N \rightarrow ...
0
votes
0answers
67 views

differentiable equal r-forms

Let $\alpha$, $\beta$ be two $r$-forms continuous in $U\subseteq \mathbb{R}^n$ open. If $\int_M \alpha =\int_M \beta $ for all surface $M\subseteq U$   dimension $r$, compact, with boundary, then ...
3
votes
1answer
79 views

de Rham cohomology of $\mathbb{R}P^n$ via action by $SO(m+1)$

In lecture, my teacher proved the theorem that given a smooth $G$-action by a compact, connected Lie group on a manifold $M$, the de Rham cohomology of the $G$-invariant differential forms $H^p_G(M)$ ...
2
votes
1answer
188 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
2
votes
1answer
131 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 ...
2
votes
1answer
371 views

Closed but not exact one-form on $S^2$

I would like to know whether there is any nice prescription to give an example of a closed but not exact one-form on $S^2$ (not the $3$-ball). I assume to take some points out of this surface, e.g. 3. ...
0
votes
1answer
37 views

Prove $\Lambda^kf:\Lambda^kW^* \to \Lambda^kV^*$ well-defined and linear

Let $V$ and $W$ be finite dimensional vector spaces, and $f : V \to W$ a homomorphism. Show that $\Lambda^kf:\Lambda^kW^* \to \Lambda^kV^*$, defined by $\Lambda^k\alpha(v_1, ..., v_k)=\alpha(f(v_1), ...
2
votes
0answers
145 views

Volume form on $\mathbb{S}^2$

Let $\omega = x_1 \,dx_2 \wedge dx_3 + x_2 \,dx_3 \wedge dx_1 + x_3\, dx_1 \wedge dx_2$, with $(x_1,x_2,x_3) \in \mathbb{S}^2$, be a volume form on $\mathbb{S}^2$, and let $f : \mathbb{S}^1 \times ...
3
votes
1answer
48 views

1-form on $S^n$ with non-degenerate differential.

How it can be solved? Whether there is a $1$-differential form $w$ on $S^n$, such that $dw$ is non-degenerate on $T_aS^n$ for each $a \in S^n$?
2
votes
0answers
54 views

Help with local coordinates computations-differential forms

I am reading a lecture notes on differential forms and tangent bundles on complex manifolds and I got stuck on one line where the author does not explain how he did the computation: Let ...
2
votes
3answers
133 views

Simple criteria for “closed $\Longrightarrow$ exact”

In determining whether a closed form is an exact form, there is a lot of differential geometry definitions etc. that come in. I'm interested: what is the dummy, Calc III version of when closed implies ...
1
vote
1answer
81 views

Visualizing Non-Zero Closed-Loop Line Integrals Via Sheets?

How do I visualize $\dfrac{xdy-ydx}{x^2+y^2}$? In other words, if I visualize a differential forms in terms of sheets: and am aware of the subtleties of this geometric interpretation as regards ...
3
votes
0answers
90 views

Definition of differential forms

I am reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. The authors introduce some definitions about differential forms at the beginning of Section 2.2. These ...
4
votes
1answer
196 views

Integral Definition of Exterior Derivative?

Is there a rigorous integral definition of the exterior derivative analogous to the way the gradient, divergence & curl in vector analysis can be defined in integral form? Furthermore can it be ...
2
votes
3answers
122 views

On the scalar product of a vector field and the exterior derivative of a smooth map on a surface

I am reading Roger Penrose's wonderful book 'The Road to Reality: A Complete Guide to the Laws of the Universe'. I need some help in understanding the scalar product of a vector field and the ...
11
votes
1answer
1k views

How to calculate the pullback of a $k$-form explicitly

I'm having trouble doing actual computations of the pullback of a $k$-form. I know that a given differentiable map $\alpha: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ induces a map $\alpha^{*}: ...
8
votes
0answers
157 views

algebraic $1$-forms vs analytic $1$-forms

First let's fix some definitions: Definitions: Complex manifold (of dimension n): Is a locally ringed space $(X,\mathscr F)$, where there is an open cover $\bigcup_{i\in I} U_i=X$ such that ...
1
vote
1answer
74 views

A question about differential forms

I have checked Rudin's proof about Poincaré Lemma (Principles of Mathematical Analysis) and it seems to have a mistake. Through Google, I found another guy who has noted such error. More details here: ...
3
votes
2answers
59 views

Poles of abelian differentials

Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$. As a corollary of the Riemann-Roch theorem we know that for every abelian differential $\omega$ on $X$ we have ...
2
votes
2answers
103 views

Show that $a \wedge * b = g(a,b) \operatorname{vol}$

$\newcommand{\vol}{\operatorname{vol}}$ Let $\omega$ be a $p$-form on a Riemannian manifold $M^n$ with metric $g$ and let $\vol_{i_1,\ldots,i_n}=\sqrt{\lvert g\rvert} \epsilon_{i_1,\ldots,i_n}$ be a ...
1
vote
1answer
40 views

Residues of a meromorphic differential on some particular points

Let $X$ be a compact Riemann surface, $\omega$ a meromorphic differential on $X$ and $f$ a meromorphic function on $X$ with poles only over the points $P_1,\dots,P_d$. The product $\;f\cdot\omega\;$ ...
7
votes
3answers
319 views

Why are differential forms more important than symmetric tensors?

In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What ...
1
vote
0answers
50 views

invariance of 2-form under $SO(3)$

I'm trying to understand how to derive forms that invariant under action of some group. For example 2-form on $S^2$ and on $\mathbb{R}^3$ is very interesting for me (because I have troubles with it). ...
2
votes
0answers
80 views

differential of a differential form

Given a differential form $w$ on a manifold, I know how to calculate $dw$ in local coordinates. But is there any way to define $dw$ independent of local coordinates?