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

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5
votes
1answer
109 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
131 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 ...
57
votes
5answers
5k views

Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
1
vote
3answers
72 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 ...
6
votes
3answers
680 views

Differential Forms and Vector Fields correspondence

Barrett O'Neill's Differential Geometry book says that Classical vector analysis avoids the use of differential forms on $\mathbb{R}^3$ by converting 1-forms and 2-forms into vector fields via the ...
1
vote
2answers
121 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
72 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
97 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 ...
0
votes
0answers
69 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 ...
4
votes
1answer
226 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
1answer
68 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 ...
4
votes
0answers
195 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
135 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 ...
3
votes
1answer
83 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
379 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
158 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
49 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 ...
1
vote
1answer
84 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 ...
2
votes
3answers
145 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 ...
3
votes
0answers
92 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 ...
2
votes
3answers
131 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 ...
12
votes
2answers
401 views

The algebraic de Rham complex

Let $A$ be a commutative $R$-algebra (or more generally a morphism of ringed spaces). Then there is an "algebraic de Rham complex" of $R$-linear maps $A=\Omega^0_{A/R} \xrightarrow{d^0} \Omega^1_{A/R} ...
10
votes
2answers
267 views

Integral of wedge product of two one forms on a Riemann surface

I'm having trouble verifying an elementary assertion made in this answer on MathOverflow. It seems more like a math.stackexchange question, so I'm asking it here. Anyway, the assertion is as follows ...
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 ...
2
votes
2answers
104 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
41 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 $\;f\cdot\omega\;$ ...
1
vote
1answer
75 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
60 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 ...
1
vote
0answers
54 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
85 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?
2
votes
2answers
148 views

p-forms as multilinear maps

I'm studying differential geometry and am learning about differential forms. We have a very intuitive and simple way to understand 1-forms as linear maps on from the tangent space to the base field, ...
1
vote
1answer
92 views

A (not so?) simple question about differential forms

Let $M^n$ be a compact orientable manifold and let $\omega$ be a $(n-1)$-form in $M^n$. I want to show that there is $p\in M$ such that $(d\omega)_p=0$. Can somebody help me, please ? Thanks :)
2
votes
0answers
34 views

Laplacian on a warped product.

Let $(M, g)$, $(N, h)$ be complete Riemannian manifolds (not necessarily compact). Let $f : M \rightarrow (0, \infty)$ be a smooth function, and finally let $$\overline{M} = M \times_f N$$ be the ...
3
votes
1answer
413 views

What's the geometrical intuition behind differential forms?

This question can look like a duplicate of this one, but it's kind of different. I'm trying to relate some geometrical meanings I've seem in some books to the definition of differential forms in ...
1
vote
0answers
232 views

$\alpha \wedge \beta = 0$ iff $\beta = \alpha \wedge \gamma$

I have been given the following problem: Let $\alpha$ be a nowhere-zero 1-form. Prove that for a (p+1)-form $\beta$ $(p\geq0)$, one has $\alpha \wedge \beta = 0$ if and only if $\beta = \alpha ...
0
votes
1answer
190 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
151 views

product of harmonic forms in a kähler manifold

In general, the product of two harmonic differential forms is not harmonic. However, for Kähler manifolds, the product of two harmonic forms is harmonic. What is a counterexample for the first ...
3
votes
2answers
313 views

wedge product with the exterior derivative of the form $ \omega:= dz +x_1 \, dy_1+ x_2 \, dy_2 + \cdots + x_n \, dy_n $.

Write the coordinates on $ \mathbb {R} ^{2n+1}$ as $ \displaystyle{ (x_1 , y_1, x_2, y_2, \cdots ,x_n, y_n ,z)}$. Define the 1-form $ \displaystyle{ \omega:= dz +x_1 \, dy_1+ x_2 \, dy_2 + \cdots + ...
0
votes
2answers
72 views

Decomposing a 2-form into a product of two 1-forms

I'm trying to decompose the 2-form $\omega = dx \wedge dy + 4dx \wedge dz + 3dz \wedge dy$ (in $\mathbb{R}^{3}$) as the product of two 1-forms, but get stuck. Is it posible to do this?
5
votes
0answers
72 views

Derivations of the algebra of differential forms

It is well known that the interior product, the Lie derivative, and the De Rham differential are derivations of the algebra of differential forms. Does there exist other derivations of this algebra ...
4
votes
2answers
225 views

Can calculus of varations be formalised with exterior calculus?

I noticed that a calculus of variations problem is just an integral over a differential form. Therefore, I would think it would be possible to formulate the Euler-Lagrange equations using exterior ...
2
votes
0answers
52 views

What is the differential of a function?

I'm reading Do Carmo's Differential Forms and Applications (1st ed) and on page 6 he takes a differentiable map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$, a point $p \in \mathbb{R}^{n}$ and a ...
1
vote
1answer
51 views

Is $\omega = dU = sin(x+y)dx+cos(x+y)dy$ an exact form?

In my thermodynamics homework I should prove that $dU = sin(x+y)dx+cos(x+y)dy$ is a function of state. Which means it's integration over any path be constant or in other word $dU$ should be an exact ...
2
votes
1answer
49 views

Cohomology of volume forms

If g and h are Riemannian metrics on the same manifold, say both of volume 1, then it follows (I guess from Poincaré duality) that their volume forms dvol_g and dvol_h are cohomologous. Question: is ...
7
votes
2answers
1k views

Cartan's magic formula

A possible proof of Cartan's magic formula $$L_X = i_X \circ d+d \circ i_X$$ is to follow the steps: Show that two derivations on $\Omega^{\bullet}(M)$ commuting with $d$ are equal iff they agree on ...
0
votes
1answer
64 views

How to prove matrix differential rules using differential forms rules

I am learning differential forms on my own through lectures on youtube and one of the things that I am attempting to do is to check if I can derive matrix differential rules using the rules for ...
5
votes
2answers
320 views

Why is arc length not a differential form?

I read that the arc length is not a differential form. But I don't understand why it isn't. I understand that differential forms are integrands and arc length is an expression which is integrable. ...
6
votes
1answer
176 views

Is $ds$ a differential form?

I am somewhat confused as to whether $ds$ (line element) is actually a differential form... we have (in $\mathbb{R}^2$): $$ds^2 = dx^2 + dy^2$$ Differential 1-forms are supposed to be linear ...