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 $1$-forms $\theta$ solve $\Delta \theta = f\theta$ for a smooth function $f$?

I have a seemingly basic question that I cannot find any literature on. Let $(M,g)$ be a smooth Riemannian manifold and let $\Delta:\Omega^1(M) \to \Omega^1(M)$ be the Laplace-De Rham operator on $1$-...
Julian Chaidez's user avatar
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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
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Friedrich's second inequality for functions with zero average

Friedrich's second inequality (or Maxwell Estimates or Gaffney’s inequality in the literature) is referred as follows: for all $\mathbf{u} \in H^1(\Omega)^2$ satisfying either $\mathbf{n} \cdot \...
Ryan Li's user avatar
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Does a gauge-invariant Caccioppoli inequality hold?

(I previously asked this question on Math.SE but got no responses after two weeks.) Let $V \Subset U$ be domains in a Riemannian manifold $M$, and $W := U \setminus \overline V$. If $u: U \to \mathbb ...
Aidan Backus's user avatar
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127 views

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 ...
Uncool's user avatar
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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 ...
Dizbro's user avatar
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1 answer
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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
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Mass of the push forward of a k-current with fixed orientation

$\DeclareMathOperator{\Mass}{Mass}$Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smoth map. Given a $2$-vector (in general a $k$-vector but let's stick to $2$) $v_1 \wedge v_2 \in \Lambda_2 (\mathbb{R}^...
tommy1996q's user avatar
1 vote
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69 views

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 $\alpha^{n-1,n-2}$ ...
Samir's user avatar
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1 answer
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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 $X$ ...
Sam Freedman's user avatar
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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 ...
Ho Man-Ho's user avatar
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1 answer
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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 ...
Frederic Thomas's user avatar
4 votes
1 answer
197 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
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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....
Sergei Ovchinnikov's user avatar
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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 ...
Taraellum's user avatar
3 votes
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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 ...
Sprotte's user avatar
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8 votes
1 answer
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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 ...
tryst with freedom's user avatar
1 vote
0 answers
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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 ...
user91126's user avatar
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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 ...
Uncool's user avatar
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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 ...
Fabrice Pardo's user avatar
1 vote
1 answer
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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 ...
Partha's user avatar
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10 votes
2 answers
743 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
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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. ...
2 votes
1 answer
571 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
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5 votes
1 answer
170 views

Finding a volume form on a fibre of a submersion between oriented manifolds

Let $f:X\to Y$ be a submersion between orientable smooth manifolds of respective dimensions $n,p$ and let $j:M=f^{-1}(y)\hookrightarrow X$ denote the inclusion of some fibre of $f$. My naïve (I am ...
Georges Elencwajg's user avatar
3 votes
1 answer
281 views

Differential forms and continuous maps

Let $$ X \xrightarrow{f} Z \xleftarrow{g} Y $$ be smooth manifolds and smooth maps (smooth = $C^\infty$), and $$ X \xrightarrow{K} Y $$ be a continuous map such that $f=g\circ K$. Let $\...
Semen Podkorytov's user avatar
11 votes
2 answers
696 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
6 votes
1 answer
163 views

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 ...
Eduardo Longa's user avatar
3 votes
1 answer
308 views

Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism

If $X\longrightarrow S$, where $S=\operatorname{Spec}(k)$, is a proper smooth morphism of dimension $p$ between $\mathbb{F}_{p}$ schemes with Frobenius $F$, then there is an exact pairing $\wedge:\...
The Thin Whistler's user avatar
1 vote
0 answers
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How does one interpret the wetting area?

This may be a simple question, but I decided to post it here (not just on MSE) because it is very related to a research topic: capillary surfaces. Let $(M^3,g)$ be a Riemannian $3$-manifold with ...
Eduardo Longa's user avatar
2 votes
1 answer
187 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
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13 votes
2 answers
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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?
Andrei's user avatar
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1 vote
0 answers
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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 $\Lambda^2(M)\to\Lambda^1(M)$. ...
Paul's user avatar
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7 votes
0 answers
214 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
  • 6,952
12 votes
2 answers
1k 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
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
5 votes
1 answer
145 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
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2 votes
1 answer
205 views

When is this differential form harmonic?

Let $(M^3, g)$ be a (closed) Riemannian manifold and let $u: M \to S$ be a harmonic function, where $S$ is a closed orientable surface. If $\omega$ is a $2$-form on $S$, what are sufficient conditions ...
Eduardo Longa's user avatar
1 vote
0 answers
162 views

Tischler's Theorem on nonvanishing $1$-forms on open manifolds

I have been trying to find a generalized version of the following theorem due to D. Tischler, Theorem 1. Let $M^n$ be a closed $n$-dimensional manifold. SUppose $M^n$ admits a nonvanishing closed $1$-...
Aaron Maroja's user avatar
8 votes
1 answer
449 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
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2 votes
0 answers
129 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
3 votes
2 answers
239 views

A question on the nature of the vortex number

In the Yang-Mills-Higgs (also called magnetic Ginzburg-Landau) model in the plane the energy has the form $$ E(A,\phi)=\int_{\mathbb{R}^2}\left(|(d-iA)\phi|^2+\frac{1}{2}F_{jk}F_{jk}+\frac{1}{4}(1-|\...
Paul's user avatar
  • 213
4 votes
2 answers
450 views

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 ...
GTA's user avatar
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9 votes
1 answer
967 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
5 votes
0 answers
105 views

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 \...
Raz Kupferman's user avatar
3 votes
1 answer
342 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
7 votes
0 answers
267 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
4 votes
1 answer
518 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
387 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 &...
roi_saumon's user avatar
5 votes
0 answers
209 views

Exact differential forms in characteristic $p>0$

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 ...
Huy Dang's user avatar
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