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Questions tagged [differential-forms]

A differential form $ \omega$ is a section of the exterior algebra $\Lambda^* T^* X$ of a cotangent bundle,

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14 votes
1 answer
1k views

When is a given matrix of two forms a curvature form?

Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of 1-forms $...
Vamsi's user avatar
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17 votes
2 answers
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Hodge decomposition in Minkowski space

This question is motivated by the physical description of magnetic monopoles. I will give the motivation, but you can also jump to the last section. Let us recall Maxwell’s equations: Given a semi-...
The User's user avatar
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37 votes
15 answers
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Geometric imagination of differential forms

In order to explain to non-experts what a vector field is, one usually describes an assignment of an arrow to each point of space. And this works quite well also when moving to manifolds, where a ...
Mircea's user avatar
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28 votes
4 answers
6k views

Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc?

My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any algorithm) ...
HYYY's user avatar
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12 votes
2 answers
852 views

Visualizing holomorphic differentials on a compact Riemann surface?

It is a classical result that the vector space of holomorphic differentials on a compact Riemann surface of genus $g$ has dimension $g$. I am wondering if there is a way of visualizing this wonderful ...
Timothy Chow's user avatar
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12 votes
2 answers
2k views

Different definitions for integral de Rham cohomology classes

Suppose that $S$ is a compact orientable surface. In this case, the top de Rham cohomology space $H^2(S)\cong \mathbb{R}$, with the isomorphism given by integration on $2$-forms along $S$. Now, one ...
G. Gallego's user avatar
12 votes
2 answers
883 views

Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$

This is a problem occurring in my research about deformations of $\mathbb{Z}/p^n$-covers over a ring of power series. Given an algebraically closed field $k$ of characteristic $p>0$, suppose $1< ...
Huy Dang's user avatar
  • 245
11 votes
2 answers
3k views

The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve

Working over the complex numbers, consider a function $F\left(x,y\right)$ and a curve $C$ defined by $F\left(x,y\right)=0$. I know that to construct the Jacobian variety associated to $C$, one ...
MCS's user avatar
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10 votes
1 answer
2k views

Hodge Laplacian in local coordinates

On a Riemannian Manifold $M^n$, the Hodge Laplacian is defined on k-forms by $\Delta\omega=\operatorname{d}\operatorname{d}^*\omega+\operatorname{d}^*\operatorname{d}\omega$. For 0-forms, e.i. smooth ...
Felix Schlag's user avatar
7 votes
0 answers
245 views

Albanese morphism induces an isomorphism on global $1$-forms

Let $X$ be a smooth projective variety over a field $k$ of characteristic zero equipped with a point $e\in X(k)$. There is Albanese morphism $a:X\to \mathrm{Alb}\,X$ which is initial among pointed ...
SashaP's user avatar
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6 votes
1 answer
404 views

Densities, pseudoforms, absolute differential forms and measures, differential forms, etc

Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here. Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
D.R.'s user avatar
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5 votes
1 answer
637 views

Leafwise de Rham cohomology (A true definition of differential forms along leaves)

For a foliated space $(M, \mathcal{F})$, one associate a leafwise de Rham cohomology. This cohomology and trace-class operators on this cohomology and trace interpretations for closed orbits of ...
Ali Taghavi's user avatar
2 votes
0 answers
313 views

Analytic version of the Cartan lemma

Assume that $\beta$ is a real analytic 2-form on an analytic manifold $M$ and $\alpha$ is an analytic non vanishing 1-form on $M$. Assume that $\beta \wedge \alpha=0$. Is there an analytic 1-form $\...
Ali Taghavi's user avatar
1 vote
2 answers
675 views

$\infty$-forms and $\infty$-plectic geometry

Can you have $\infty$-forms on infinite-dimensional manifolds or elsewhere and what are they used for?
user17731's user avatar