Articles of differential forms

Problem with $Pullback$ Calculation

Let $$\omega=-xdx \wedge dy-3dy\wedge dz$$ and $$\phi:\mathbb R^2 \to \mathbb R^3, (u,v) \to(uv,u^2,3u+v)$$ I tried to compute the pullback $\phi^*(\omega)$, but was not able to solve it. $$\phi^*(\omega)=(-uv)(vdu+udv)-3(2udu)\wedge(3du+dv)=-uv^2du-u^2vdv-6udu\wedge(3du+dv)=uv^2du-u^2vdv-6udu\wedge3du-6udu\wedge dv$$ Could someone tell me, what I am doing wrong?

Lie derivative of a covector field

The lecturer here wants the viewer to derive the components of the Lie derivative of a (1,1) tensor-field. To this end, I want to derive the components of the Lie derivative of a covector field: let $(U,x)$ be a chart, $\omega, \chi$ covector-fields and $X,Y$ vector fields on the smooth manifold $(M,\mathcal{O},\mathcal{A})$, I get: I […]

Any two $1$-forms $\alpha$, $\alpha'$ with property satisfy $\alpha = f\alpha'$ for some smooth nowhere zero function $f$?

This is a followup question to here. Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. Is there a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a section of $\xi$? Does it follow that any two $1$-forms $\alpha$, $\alpha’$ […]

Confusion about holomorphic differential on elliptic curve

Let $(a:b)\in\mathbb{C}P^1$ and look at the elliptic curve $C$ given by $y^2=x^3+a^4x+b^6$. It is well known that on this elliptic curve we have the holomorphic differential $dx/y$. I have two questions about this: I actually don’t understand very well why $dx/y$ is a holomorphic differential. What happens when $y=0$? If you have a reference where […]

How can I prove $dz=dx+idy$?

Let’s see $\Bbb C$ as an $\Bbb R$-vector space. Hence it is isomorphic to $\Bbb R^2$ and it has dimension $2$. If $v_1,v_2$ is a basis for $\Bbb R^2$, every its element can be written as $xv_1+yv_2$; in coordinates $(x,y)$. Let’s consider now the dual space $(\Bbb R^2)^{*}:=\operatorname{Hom}_{\Bbb R}(\Bbb R^2,\Bbb R)$; a base for it […]

Generators of $H^1(T)$

Let $T$ denote the torus. I am working towards an understanding of de Rham cohomology. I previously worked on finding generators for $H^1(\mathbb R^2 – \{(0,0)\})$ but then realised that for better understanding I had to look at different examples, too. For the purpose of this question I am only interested in finding just one […]

Exterior derivative of a 2-form

I want to prove that the exterior derivative of a 2-form in $\mathbb{R}^n$: $${\alpha = \sum a_{ij} dx_i \wedge dx_j}$$ is: $${d \alpha = \sum (\frac{\partial a_{ij}}{\partial x_k} + \frac{\partial a_{jk}}{\partial x_i} – \frac{\partial a_{ik}}{\partial x_j})dx_i \wedge dx_j \wedge dx_k.}$$ The thing is that I’m stucked after taking the differential of the function $a_{ij}$ and […]

Trouble understanding proof of this proposition on contact type hypersurfaces

I have finally started working on my thesis. I am reading the first reference book, which is Dusa McDuff, D. Salamon, Introduction to Symplectic Topology. It’s a tough struggle, given my not-too-great experience with differential forms. I will recall a few concepts. A manifold $M$ is called symplectic if it is equipped with a closed […]

Relation between de Rham cohomology and integration

This question is a follow-up to When does a null integral implies that a form is exact? . As mentionned in the selected answer, given certain conditions it is possible to find an isomorphism between the top de Rham cohomology and the integral $\int_M \omega$ of the members of each equivalence class $\omega$. Moving on […]

Zeroes of exact differential forms on compact manifold

Let $M$ be a $n$ dimensional compact differentiable manifold. I would like to show that any exact differential form of degree $n$ vanishes at at least one point. I think it is a generalization of the following fact : if $f$ is a diffferentiable function on $M$ then it either has a maximum or is […]