# 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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### What is the topology on the space of differential forms $\Omega^2(M)$?

I have posted this question on MSE one year ago, but till now I did not received an answer. Therefore I have decided to post it here. I have difficulty in understanding the meaning of "A ...
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### Degeneration differential form nodal curve

I have a (possibly very basic) question about differential forms on nodal curves. After reading Witten's survey "Two-dimensional gravity and intersection theory on moduli space", I am ...
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### Vanishing components of Kähler metric

Let $(X, \omega)$ be a $n$-dimensional complex Kähler manifold such that $\omega^{n-1}=d\alpha$. Does $\partial\alpha^{n-1,n-2} =0$ (resp. $\bar\partial\alpha^{n-2,n-1} =0$) Where $\alpha^{n-1,n-2}$ ...
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### Detecting non-affine automorphisms of a translation surface

Let $(X, \omega)$ be a translation surface, i.e., a Riemann surface with a homomorphism $1$-form. A central object is the group of affine automorphisms $\text{Aff}^+(X, \omega)$: homeomorphisms of $X$ ...
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### Proving an equality of differential forms by assuming some perhaps topological condition

Let say I want to show two differential forms $\omega_1$ and $\omega_2$ on a smooth manifold $M$ are equal. Of course it suffices to show $\omega_1=\omega_2$ locally, i.e. the equality holds over ...
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### Behaviour of the Cartan Maurer form

Let there be a Lie-group $G$ and its Lie-algebra $g$. Then the Cartan Maurer form is an 1-form $\omega: T_gG \rightarrow T_eG$ for which holds: $$(L^\ast_g)\omega = \omega$$ In Shlomo Sternberg's ...
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### 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 ...
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### 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....
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### 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 ...
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### 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 ...
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### 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 ...
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### Target space of Green's operator on $L^p$-differential forms on closed manifolds

Let $M$ be a closed (i.e., compact without boundary) smooth oriented Riemannian manifold endowed with a regular atlas in the sense of C. Scott [1], i.e., with a finite atlas $\mathcal{A}$ so that for ...
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### $\omega$ is a symplectic form then $L^k_{\omega}:\Omega^q(M)\to \Omega^{q+2k}(M)$ is injective for all $q\leq n-k$ and surjective for all $q\geq n-k$

Let $(M^{2n},\omega)$ be a symplectic manifold of dimension $2n$. Let $L^k_{\omega}:\Omega^q(M)\to \Omega^{q+2k}(M)$ be the map given by $L^k_{\omega}(\alpha)=\alpha\wedge\omega^k$. Then is it true ...
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### Decomposition of a bivector of a Lorentzian manifold [closed]

In the case of the $(1, -1, -1, -1)$ metric, a bivector (or a 2-form) $F$ decomposes uniquely as the sum of two orthogonal simple bivectors if $F^2 \ne 0$. I have the impression that it is very little ...
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### A curve in the bundle of two forms

Let $(M,g)$ be a closed Riemannian manifold. Fix a point $m\in M$ and a $2$-form $\omega$ at $m.$ Take a curve $\gamma$ in $M$ such that $\gamma(0)=m.$ Now we can get a $2$-form along $\gamma$ by ...
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### 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, ...
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### 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. ...
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### 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 ...
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### Tangential harmonic $1$-forms are pullbacks of harmonic functions

This question has also been posted on MSE, but maybe here is the right place to obtain an answer. Let $(M^3,g)$ be a compact connected oriented Riemannian $3$-manifold with nonempty boundary. The ...
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### 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?
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### Codifferential of wedge of two 1-forms

Let $\omega,\eta$ be two 1-forms on a manifold $M$. I'm interested in an expression for $$\delta(\omega\wedge\eta)$$ where $\delta$ is the co-differential operator $\Lambda^2(M)\to\Lambda^1(M)$. ...
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### 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 ...
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### 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 ...
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### Kinds of differentials and algebraic groups

This Wikipedia article mentions that the analogues of differentials of the first/second/third kind for algebraic groups are abelian varieties/algebraic tori/linear algebraic groups. I guess ...
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### 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 ...
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### On boundary-value problems for differential forms on a manifold

Let $M$ be a simply-connected $d$-dimensional Riemannian manifold with boundary (for simplicity assume a ball). Consider the boundary value problem for $\omega\in\Omega^k(M)$,  d\omega = \alpha \...
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### 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 ...
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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 $\beta(Y)=\langle Y,X\... • 270 4 votes 1 answer 481 views ### Leafwise de Rham cohomology (A true definition of differential forms along leaves) For a foliated space$(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 ... • 270 2 votes 0 answers 376 views ### Why is the integral of the tautological 1-form equal to the action? I am having a hard time to understand why the integral of the tautological 1-form is the action of the system. The tautological one form is defined by : \begin{align} \theta_{(q,p)} : T_{(q,p)}T^*Q &... 5 votes 0 answers 206 views ### Exact differential forms in characteristicp>0$Let$k$be an algebraically closed field of characteristic$p>0$. Suppose$1< e_i <p$for$i=1,2, \ldots, n$are integers ($n \ge 2$). What are the conditions on the$e_i$'s so that the ... • 245 2 votes 0 answers 107 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}....
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 ...