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37 votes
15 answers
13k views

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
  • 2,041
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
  • 1,499
25 votes
3 answers
5k views

Integration and Stokes' theorem for vector bundle-valued differential forms?

Is there a version of Stokes' theorem for vector bundle-valued (or just vector-valued) differential forms? Concretely: Let $E \rightarrow M$ be a smooth vector bundle over an $n$-manifold $M$ equipped ...
gspr's user avatar
  • 465
17 votes
2 answers
4k views

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
  • 2,442
15 votes
2 answers
888 views

Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?

Fix a compact Riemannian manifold $M$ (leaving the metric implicit). What I'd like to know is if the corresponding Hodge decomposition of smooth $n$-forms $$ \Omega^n(M) \simeq \mathcal{H}^n(M)\oplus ...
David Roberts's user avatar
  • 35.5k
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
  • 3,383
14 votes
0 answers
573 views

Reference for a proof of the fiberwise Stokes theorem

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary, the difference between the fiberwise integral of the differential and the ...
Dmitri Pavlov's user avatar
13 votes
3 answers
2k views

k-form: sum of wedge products of 1-forms?

Let M be a smooth manifold. Can every k-form $\omega$ on M be written as a sum of k-forms, that are wedge products of 1-forms, i.e. $\omega = \sum_{i=0}^n \alpha_1^{(i)} \wedge \ldots \wedge \alpha_k^{...
jsb's user avatar
  • 403
13 votes
3 answers
784 views

Volume-minimizing submanifold implies calibrated?

Let $X$ be a smooth manifold of dimension $d$ and $M$ an oriented submanifold of dimension $p < d$ so that the multiples k⋅M are absolutely minimizing $p$-volume in their integral homology classes ...
Michael Freedman's user avatar
13 votes
2 answers
1k views

Poynting vector and differential forms

It is well known that electromagnetic field is a 2-form and Maxwell's equation can be reformulated in language of differential forms. What is the Poynting vector in this language?
Andrei's user avatar
  • 139
11 votes
0 answers
354 views

Which differential forms commute with the curvature form?

Consider a vector bundle, $E \to M$, with connection, $\nabla$, and curvature $2$-form, $F$ on $M$. For $E$-valued differential forms on $M$, $\Omega(M, E)$, we have an exterior covariant derivative, ...
cheyne's user avatar
  • 1,611
10 votes
2 answers
891 views

Sobolev spaces of differential forms and regular atlases

In [1] (section 3), C. Scott introduces the following concept of regular atlas for closed $C^\infty$-smooth Riemannian manifolds. He says: When referring to a coordinate system $(U,\phi)$ as regular, ...
user91126's user avatar
  • 213
10 votes
2 answers
1k views

Odd differential forms

In de Rham's classical book "Variétés Différentiables" de Rham, Georges, Variétés différentiables. Formes, courants, formes harmoniques. 3e éd. revue et augmentée, Publications de l’Institut de ...
Alan Muniz's user avatar
8 votes
1 answer
2k views

What do the differential k-forms on a product manifold look like?

I am interested in how I could express $\Omega^k( M \times N)$ in terms of $\Omega^i(M)$ and $\Omega^j(N)$ for $i,j = 0,1, \ldots k$. Is there a nice relation? This question arose in the context of ...
Todd N's user avatar
  • 313
8 votes
1 answer
864 views

Geometric definition of divergence using curvature mentioned in Tristan Needham

In page-479 of Visual Complex Analysis, Tristan Needham derives the flux of a vector field in Geometric form: $$ \nabla \cdot X = \partial_s |X| + \kappa_p |X|$$ The $\partial_s$ is a derivative along ...
Brian's user avatar
  • 1,525
8 votes
1 answer
328 views

Condition on a differential form arising from the theory of elasticity

Let $D$ be the unit $n$-ball (for concreteness). Let $\beta\in\Omega^1(D;R^n)$ be an $R^n$-valued one-form, having full rank (viewed as a section of $T^*D\otimes R^n$). Under what conditions on $\beta$...
Raz Kupferman's user avatar
8 votes
1 answer
480 views

Differential forms on standard simplices via Whitney extension vs diffeological structure

The standard simplices $\Delta^n \subset \{\mathbf{x}\in\mathbb{R}^{n+1}\mid x_0 + \ldots + x_n =1 \} =: \mathbb{A}^n$ carry two natural sorts of smooth differential forms: Those differential forms ...
David Roberts's user avatar
  • 35.5k
7 votes
7 answers
503 views

Theorems similar to Tischler fibering theorem

Tischler theorem states that the existence of a nowhere vanishing closed $1$-form in a compact manifold $M$ implies that the manifold fibers over $S^1$. Do you know any other differential topology ...
Reb's user avatar
  • 261
7 votes
0 answers
282 views

A cohomology associated to a vector field on a Riemannian manifold

Edit: Accoring to the comment of Asura Path I revise the question. Let $X$ be a vector field on a Riemannian manifold $(M,g)$. So we have a $1$-form $\beta$ with $\beta(Y)=\langle Y,X\...
Ali Taghavi's user avatar
6 votes
1 answer
291 views

Strange problem about triplets of differential forms

Suppose we have the following map: $$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$ $$(\alpha,\beta,\gamma)\longmapsto(\mathrm{d}\alpha+\beta\wedge\gamma,\mathrm{d}\beta+\...
Jjm's user avatar
  • 2,091
6 votes
1 answer
400 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
  • 831
6 votes
1 answer
163 views

Restriction of "Spin(7) 4-form" to $\mathbb{R}_+\times S^7$

I asked this question on stackexchange (https://math.stackexchange.com/questions/2212226/restriction-of-spin7-4-form-to-mathbbr-times-s7) but was advised to ask again here: I'm currently reading ...
Martin van Beek's user avatar
5 votes
3 answers
1k views

Non-continuous differentiability for differential forms

Generally when working with differential forms, one assumes that they are continuously differentiable, i.e. $C^r$ for some $1\le r \le \infty$. Under this hypothesis, one can define the exterior ...
Mike Shulman's user avatar
  • 66.8k
5 votes
1 answer
147 views

Equivalence generated by Jacobian minors

Let $f,g:\mathbb{R}^m \to \mathbb{R}^n$ be two smooth functions and let $k$ be a strictly positive integer. Write $f \sim_k g$ if at each point in the domain, the determinants of all $k \times k$ ...
Vidit Nanda's user avatar
  • 15.5k
5 votes
1 answer
634 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
5 votes
0 answers
93 views

Is the pullback of differential forms on a compact manifold smooth tame as a map of Fréchet manifolds?

In Hamilton's paper on the Nash-Moser inverse function theorem he shows that if $M$ is a smooth compact manifold and $V\to M$ a smooth vector bundle then its smooth sections $\Gamma(V)$ equipped with ...
Jan Heck's user avatar
5 votes
0 answers
82 views

Interpolating from a Hard Lefschetz class to a Kaehler class

Let $X$ be a compact smooth manifold that admits symplectic and Kaehler structures. There is a paper by Ugarte, Rudyak, Tralle, and Ibanez, showing how the Lefschetz rank can vary along a path of ...
Sinister Cutlass's user avatar
4 votes
1 answer
455 views

Closed $3$-manifold, $2$-dimensional subbundle of this manifold, is this form exact or not?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. I know the following. There is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector ...
user avatar
4 votes
1 answer
248 views

Cohomology of invariant differential forms

Let $M$ be a compact manifold and $\varphi:M\rightarrow M$ a diffeomorphism. The invariant differential forms $$ \Omega^{k}_{inv}(M)=\{\alpha\in\Omega^{k}(M):\varphi^{*}\alpha=\alpha\} $$ form a ...
studiosus's user avatar
  • 305
4 votes
1 answer
177 views

Exact forms, gauge transformations, and the Hodge decomposition in non-abelian Gauge theory

I am trying to understand how the Hodge decomposition is affected by gauge transformations in non-abelian in gauge theory (eg $\mathrm{SU}(N)$). In particular, I am searching for a way to generalise ...
b0bgary's user avatar
  • 41
4 votes
1 answer
1k views

Norm of a differential form [closed]

How can we explicitly calculate the norm of a differential form? For example let $(X, \omega) $ be a complex manifold such that locally $$ \omega(z) =i\sum_{k,j} h_{k, j} (z) dz_k\wedge d\overline {...
user161399's user avatar
3 votes
6 answers
2k views

The purpose of connections in differential geometry [closed]

I am currently reading through differential geometry as a mathematics graduate. Can somebody give me a brief explainer on the purpose of connections? I could also use explainers on differential forms. ...
3 votes
1 answer
390 views

One-Forms in Functional Space?

I am presently reading this paper on covariant phase space and I have difficulty understanding the following formalism developed: In the paper (section $2.2$, pg. $12$), the authors have introduced ...
Spoilt Milk's user avatar
3 votes
0 answers
100 views

Understanding the Lie derivative by multivector fields

For a vector field $X$ on a manifold there are two ways to define a Lie derivative: an algebraic one using Cartan's formula $\mathcal{L}_X \alpha = i_X d \alpha + d i_X \alpha$ and a dynamical one ...
mlainz's user avatar
  • 161
3 votes
0 answers
67 views

Combinatorial approximation to the integral of a form?

This is a bit of a followup to my previous question Intuition for the volume form - combinatorial definition?. I am looking for a certain combinatorial intuition when it comes to integrating ...
Sprotte's user avatar
  • 1,075
3 votes
0 answers
298 views

Is this a 2-cyclic cocycle ? Does it have a nontrivial geometric interpretation?

Let $S$ be a surface in $\mathbb{R}^3$. Inspired by the $2$-cyclic cocycle defined in page 20 of the book "Non-commutative geometry" by Alain Connes, we consider the following $3$-linear ...
Ali Taghavi's user avatar
3 votes
0 answers
1k views

About Frobenius's theorem for differential forms

My question is about a particular case of Frobenius's theorem that states the complete integrability condition for a Pfaff system. Namely, when dealing with a system reduced to a single 1-form, the ...
jacaboul's user avatar
  • 327
3 votes
0 answers
304 views

Differential ideals of Pfaffian forms on jet bundles (Integrability)

(I asked this question on math.stackexchange, but got no reaction in several weeks. So, my conclusion is, that it is harder to answer than I thought, and maybe admissible for the attribute 'research ...
cknoll's user avatar
  • 203
2 votes
1 answer
676 views

Why non closed differential forms do not play important role for the topology of a manifold?

Cross-posted from MSE. I know that De Rham cohomology reveal some properties of the topology of smooth manifolds by finding closed differential $k$-forms $\mathsf{d}\omega=0$ that are not exact $\...
C.F.G's user avatar
  • 4,195
2 votes
2 answers
655 views

About hypersurfaces in R^n+1 with bounded 2nd fundamental form

Assume $M^n$ is a compact hypersurface without boundary immersed in $R^{n+1}$, with $A$ its 2nd fundamental form. If the square norm of A is bounded by an abstract constant, i.e. $|A|^2\leq C$ for ...
Li Yan's user avatar
  • 21
2 votes
1 answer
200 views

Vanishing product of a closed and coclosed form on a Riemannian manifold

For a (compact) Riemannian manifold $(M,g)$, can it happen that for a non-zero form $\text{d}^*\omega$, and a smooth function $f$ such that $\text{d}f \neq 0$, we can have $$ \text{d}f \wedge \text{d}^...
Max Schattman's user avatar
2 votes
1 answer
391 views

(n-1)-dimensional normal currents and Smirnov's paper

I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper http://www.unige.ch/~...
A random mathematician's user avatar
2 votes
1 answer
199 views

Decomposition of forms on a Spin$(7)$ manifold

Let's take a $G_2$ manifold $(M,\Phi)$, then we get a Spin$(7)$ manifold by taking $(M\times\mathbb{R},\Psi:=\Phi\wedge dt+*_M\Phi),$ where $t$ is the coordinate in the $\mathbb{R}$-direction. $\Phi\...
Partha's user avatar
  • 954
2 votes
1 answer
193 views

Why does the formula (7) in the article equivariant cohomology with generalized coefficients hold

I'm reading the article of equivariant cohomology with generalized coefficients by Kumar and Vergne and I have this question from that article Let G be a compact lie group with lie algebra $\mathfrak{...
Mira's user avatar
  • 139
2 votes
0 answers
65 views

Lefschetz operator on bundle-valued forms

For a holomorphic vector bundle $V \rightarrow X$ endowed with a Hermitian structure, one may define the corresponding Dolbeault-like operators $\bar{\partial}_V: \Omega^{p,q}(V) \rightarrow \Omega^{p,...
Eweler's user avatar
  • 121
2 votes
0 answers
134 views

Derivative of anti-self-dual forms on Kähler space

I am puzzled if we can establish differential relations about anti-self-dual 2-forms on the Kähler space similar to ones for self-dual forms? Let $(\mathcal{M},g,J,\omega = J^{(1)})$ be a Kähler space....
Sergei Ovchinnikov's user avatar
2 votes
0 answers
134 views

Norm of the Lipschitz-Killing differential forms

I am currently learning about the theory of Normal Cycles which makes use of the language of currents and differential forms. They are defined in the following way The Lipschitz-Killing curvature form ...
Taraellum's user avatar
2 votes
0 answers
136 views

Heat-Flow on continuous differential forms and the Feller peroperty

Let (M,g) be a complete Riemannian manifold. It is well known that the Laplace operator is essentially self-adjoint on $C^\infty_c(M)$. This extends to the (de Rahm) Laplace operator on forms. Thus in ...
Nathanael Schilling's user avatar
2 votes
0 answers
113 views

Computation of equivariant 3 form

I want to how an equivariant 2-form and equivariant 3- form look like i,e., Let M be a complex Manifold say $ S^4 \subset C^3$ and a compact lie group $S^1$ acting on it via the action $\exp{i\theta}....
Anantadulal paul'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