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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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 ...
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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 ...
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Integrate unit normal vector over unit sphere intersected with a simplicial cone
Let $S^{d-1}$ be the unit sphere in $\mathbb R^d$. Consider a ($d$-dimensional) simplicial cone $C$ in $\mathbb R^d$ whose extremal rays are spanned by some unit vectors $\mathbf{u}_1,\ldots,\mathbf{u}...
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Notion of Kahler differentials for Berkovic spaces
What is, in abstract analytic geometry (I mean, for example, in Berkovic spaces), the approach used for differential forms?
Ordinary Kahler differentials from commutative algebra/algebraic geometry ...
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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 ...
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Availability of a copy the first volume of Segre's "Forme differenziali e loro integrali"
I am precisely referring to the following, first volume of the textbook/lecture notes/monograph written by Beniamino Segre in the fifties of the twentieth century (I own a copy of the second volume)
...
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The wedge product of two positive forms is positive
I have previously posted this question on MSE, but still didn't solve it.
Definition. A real $(p, p)$-form $\psi$ on a complex manifold $M^{n}$ is said to be (semi-) positive, if for any $x \in M$, ...
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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 ...
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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 \...
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Projection to trivial reduced cohomology class in $L^2(\mathbb{R})$
Given that I have had no success on the mathematics stackexchange (see here), I've decided to try my luck here.
I am attempting to solve the following exercise (original formulation here), which to my ...
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Sign of the permutation when I show that $\star{\star w}= (-1)^{n(n-k)} w$ for the Hodge operator
Let $w=\sum_{I} a_{I}dx_{I}$ be a $k$-form in $\Bbb R ^n$. Let us consider the Hodge operator in a combinatorial form, i.e. as an $(n-k)$ form such that
$$\star(dx_{i_{1}} \wedge \dotsb \wedge dx_{i_{...
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Application of $k$-forms to differential equations
I was presented with a problem that, in my view, is somewhat difficult, and it relates $k$-forms to systems of differential equations.
Consider the ellipsoid, given by
$$f(u, v)=(a\sin(u) \cos(v), b \...
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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 ...
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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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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 ...
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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$-...
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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,...
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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 ...
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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) ...
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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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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\...
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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}^...
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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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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 ...
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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$$
In Shlomo Sternberg's ...
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Integration and Stokes' theorem for vector bundle-valued differential forms?
Is there a version of Stokes' theorem for vector bundle-valued (or just vector-valued) differential forms?
Concretely: Let $E \rightarrow M$ be a smooth vector bundle over an $n$-manifold $M$ equipped ...
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Cohomology of invariant differential forms
Let $M$ be a compact manifold and $\varphi:M\rightarrow M$ a diffeomorphism. The invariant differential forms
$$
\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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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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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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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 ...
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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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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 $\...
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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 ...
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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 $\...
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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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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:\...
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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 ...
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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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