Questions tagged [differential-forms]

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

Filter by
Sorted by
Tagged with
4 votes
1 answer
136 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 ...
  • 265
2 votes
0 answers
80 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....
0 votes
0 answers
101 views

For a vector field $f$ and a measure $\mu$, does conservativity $\mu$-almost everywhere have a sense?

$\DeclareMathOperator\Jac{Jac}$ Let $\Omega$ an open star shaped subset of $\mathbb{R}^d$ and $f : \Omega \rightarrow \mathbb{R}^d$ a differentiable vector field. For $x \in \mathbb{R}^d$, we denote ...
2 votes
0 answers
41 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 ...
0 votes
0 answers
78 views

Kähler cone of a product $K3\times K3$

Let $X$ and $Y$ be K3 surfaces, with Kähler cones $K_X$ and $K_Y$. If $\omega_1,\omega_2$ are Kähler forms on $X,Y$ respectively, $\omega=\pi_X^*\omega_1+\pi_Y^*\omega_2$ is a Kähler form on $X\times ...
  • 141
3 votes
0 answers
59 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 ...
  • 757
8 votes
1 answer
581 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 ...
1 vote
0 answers
75 views

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 ...
  • 213
2 votes
1 answer
203 views

$\omega$ 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 ...
  • 161
10 votes
0 answers
733 views

De Giorgi's structure theorem and Federer's characterization theorem in higher codimension

We say that a $\mathscr{H}^{m}$-measurable set $M\subset\mathbb{R}^n$ is $m$-rectifiable if there are countably many Lipschitz maps $f_j:\mathbb{R}^m\to \mathbb{R}^n$ and a $\mathscr{H}^m$-null set $...
  • 701
2 votes
0 answers
54 views

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 ...
1 vote
1 answer
77 views

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 ...
  • 515
10 votes
2 answers
553 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, ...
  • 213
3 votes
6 answers
1k 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
397 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 $\...
  • 4,085
5 votes
1 answer
133 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 ...
3 votes
1 answer
236 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 $\...
9 votes
2 answers
471 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 ...
6 votes
1 answer
133 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 ...
3 votes
1 answer
255 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:\...
1 vote
0 answers
70 views

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 ...
2 votes
1 answer
178 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{...
  • 99
13 votes
2 answers
883 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?
  • 139
0 votes
0 answers
206 views

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 $\Lambda^2(M)\to\Lambda^1(M)$. ...
  • 203
6 votes
0 answers
154 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 ...
  • 6,677
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 ...
4 votes
1 answer
659 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 {...
5 votes
1 answer
137 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 $k \times k$ ...
2 votes
1 answer
156 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 ...
1 vote
0 answers
102 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 $1$-...
7 votes
1 answer
390 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 ...
  • 31.8k
2 votes
0 answers
109 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 ...
3 votes
2 answers
222 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-|\...
  • 203
4 votes
2 answers
417 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 ...
  • 880
8 votes
1 answer
771 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 ...
5 votes
0 answers
86 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 \...
3 votes
1 answer
296 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 ...
7 votes
0 answers
244 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\...
4 votes
1 answer
398 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 ...
2 votes
0 answers
350 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
201 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 ...
  • 245
2 votes
0 answers
102 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}....
10 votes
2 answers
2k 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 ...
  • 1,216
2 votes
1 answer
156 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}^...
2 votes
1 answer
158 views

Extended Abel-Jacobi theorem

Let $X$ be a compact Riemann surface, $u$ a meromorphic function on $X$ with divisor supported on a set of points $\{P_1, ..., P_n\}$, and $f$ a meromorphic function on $X$ such that $f$ has no pole ...
3 votes
0 answers
297 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 ...
5 votes
2 answers
288 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 diffential topology ...
  • 233
4 votes
1 answer
443 views

Computing relative cohomology class of differential form

When dealing with a top degree differential form $\mu$ in a manifold $M$, a way of "computing" its cohomology class is integrating it through the whole manifold. For instance, if the integral $ \int_M ...
  • 233
12 votes
2 answers
854 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< ...
  • 245
8 votes
1 answer
284 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 $\beta$...