Questions tagged [differential-forms]
A differential form $ \omega$ is a section of the exterior algebra $\Lambda^* T^* X$ of a cotangent bundle,
40 questions with no upvoted or accepted answers
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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 ...
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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, ...
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Countability assumption for good covers in Bott-Tu
In chapter II of their text Differential Forms in Algebraic Topology, Bott and Tu construct the Čech-De Rham complex with regards to an open covering indexed by some ordered and countable indexing set....
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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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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\...
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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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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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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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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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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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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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 ...
2
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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 &...
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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}....
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Does the sheaf of locally exact differential forms splitting in positive characteristic
Let k be an algebraically closed field of characteristic $p>2$, $X$ a smooth projective curve of genus $g>1$ over $k$, and $F_X:X\rightarrow X$ be the absolute Frobenius morphism. Let $B^1_X$ be ...
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Wedge product of entries of a matrix & Volume form of the Siegel metric
Let $A=(a_{ij})$ be an $n\times n$ square matrix, and $\omega(A)=\bigwedge\limits_{i,j=1}^na_{ij}$ be the wedge product of its entries. Then, if $B=UA=(b_{ij})$ for some square matrix $U$, I think one ...
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Analytic version of the Cartan lemma
Assume that $\beta$ is a real analytic 2-form on an analytic manifold $M$ and $\alpha$ is an analytic non vanishing 1-form on $M$. Assume that $\beta \wedge \alpha=0$. Is there an analytic 1-form $\...
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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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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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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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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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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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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 ...
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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)
$$
where $\delta$ is the co-differential operator $\Lambda^2(M)\to\Lambda^1(M)$. ...
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0
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214
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Tischler's Theorem on nonvanishing $1$-forms on open manifolds
I have been trying to find a generalized version of the following theorem due to D. Tischler,
Theorem 1. Let $M^n$ be a closed $n$-dimensional manifold. SUppose $M^n$ admits a nonvanishing closed $1$-...
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differential forms in double field theory
In double field theory, there are 'double differential forms' meaning that the standard 1-forms $d x^i$ generate an algebra over functions depending on both of the double coordinates
$x^i$ and $\tilde ...
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superdiff forms and tensors
Where is it written that symmetric tensors (i.e. with multiindices)
occur as the coefficient functions of super differential forms
or rather odd differential forms?
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128
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Notion of Kahler differentials for Berkovic spaces
What is, in abstract analytic geometry (I mean, for example, in Berkovic spaces), the approach used for differential forms?
Ordinary Kahler differentials from commutative algebra/algebraic geometry ...
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84
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Application of $k$-forms to differential equations
I was presented with a problem that, in my view, is somewhat difficult, and it relates $k$-forms to systems of differential equations.
Consider the ellipsoid, given by
$$f(u, v)=(a\sin(u) \cos(v), b \...
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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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265
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Explicit adjunction formula and local top form
I am reading the section 4.2 in Kollar-Mori, where they provide the explicit equations for Du Val Singularities. In the course of the proof, they reduce to studying the equation $x^2+f(x,y)=0$ in a ...
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