Questions tagged [differential-forms]
A differential form $ \omega$ is a section of the exterior algebra $\Lambda^* T^* X$ of a cotangent bundle,
112 questions
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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
$$
\omega(z) =i\sum_{k,j} h_{k, j} (z) dz_k\wedge d\overline {...
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6
answers
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The purpose of connections in differential geometry [closed]
I am currently reading through differential geometry as a mathematics graduate.
Can somebody give me a brief explainer on the purpose of connections?
I could also use explainers on differential forms. ...
3
votes
2
answers
249
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A question on the nature of the vortex number
In the Yang-Mills-Higgs (also called magnetic Ginzburg-Landau) model in the plane the energy has the form
$$
E(A,\phi)=\int_{\mathbb{R}^2}\left(|(d-iA)\phi|^2+\frac{1}{2}F_{jk}F_{jk}+\frac{1}{4}(1-|\...
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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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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 $\...
- 809
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1
answer
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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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0
answers
105
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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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3
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0
answers
67
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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 ...
- 1,075
3
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0
answers
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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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0
answers
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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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votes
1
answer
678
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 $\...
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votes
2
answers
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About hypersurfaces in R^n+1 with bounded 2nd fundamental form
Assume $M^n$ is a compact hypersurface without boundary immersed in $R^{n+1}$, with $A$ its 2nd fundamental form.
If the square norm of A is bounded by an abstract constant, i.e. $|A|^2\leq C$ for ...
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answer
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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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votes
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answer
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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
$$
\text{d}f \wedge \text{d}^...
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1
answer
245
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Is there a matrix that converts the gradient of every possible function to gradient of other function?
I have already asked this question on math.stackexchange.com
https://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe
Now I ...
- 131
2
votes
1
answer
170
views
Extended Abel-Jacobi theorem
Let $X$ be a compact Riemann surface, $u$ a meromorphic function on $X$ with divisor supported on a set of points $\{P_1, ..., P_n\}$, and $f$ a meromorphic function on $X$ such that $f$ has no pole ...
- 1,109
2
votes
1
answer
391
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(n-1)-dimensional normal currents and Smirnov's paper
I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper
http://www.unige.ch/~...
2
votes
1
answer
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Homology of a region of the plane
This is related to this MO question, I don't know if it's really "research-level". As in that question, let $U$ be a domain of the complex plane $\mathbb{C}$, i.e. an open connected subset. Let
$$ \...
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1
answer
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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 $2$) $v_1 \wedge v_2 \in \Lambda_2 (\mathbb{R}^...
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Decomposition of forms on a Spin$(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\...
- 954
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votes
1
answer
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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{...
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2
votes
1
answer
233
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 ...
- 2,095
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votes
1
answer
102
views
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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2
votes
0
answers
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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 $1$-...
- 1,231
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0
answers
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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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0
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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 ...
- 191
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votes
0
answers
135
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....
2
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0
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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 ...
- 93
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votes
0
answers
64
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 ...
2
votes
0
answers
137
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 ...
2
votes
0
answers
425
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 &...
- 71
2
votes
0
answers
113
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}....
2
votes
0
answers
286
views
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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votes
0
answers
616
views
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 ...
- 21
2
votes
0
answers
313
views
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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vote
2
answers
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views
$\infty$-forms and $\infty$-plectic geometry
Can you have $\infty$-forms on infinite-dimensional manifolds or elsewhere and what are they used for?
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vote
1
answer
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views
Can every De Rham cohomology class be represented by a closed form $\alpha$ with $L_X \alpha=0$
Assume that $M$ is a manifold and $X$ is a vector field on $M$.
Is it true to say that every closed form is De Rham-cohomologue to a closed form $\alpha$ with $L_X \alpha =0$?
- 366
1
vote
1
answer
200
views
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
$$\star(dx_{i_{1}} \wedge \dotsb \wedge dx_{i_{...
- 251
1
vote
2
answers
127
views
Integrability at $z$ of the 2-form $ d\omega=\frac{\partial_{\bar{\zeta}}g(\zeta)}{\zeta-z}d\zeta\wedge d\bar{\zeta} $
Given $g\in\mathcal{C}^1(\bar\Delta)$, and $z\in\Delta$, how can i prove that the 2-form
$$
d\omega=\frac{\partial_{\bar{\zeta}}g(\zeta)}{\zeta-z}d\zeta\wedge d\bar{\zeta}
$$
is integrable in $z$?
At ...
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1
vote
1
answer
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views
existence of meromorphic differentials with non vanishing residues
Let $X$ be a smooth projective variety over a field $k$ of characteristic zero and let $D$ be a simple normal crossing divisor on $X$, with irreducible components $D_i$.
Does there exist a nonzero ...
- 11
1
vote
1
answer
125
views
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 \...
- 31
1
vote
1
answer
90
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 ...
- 954
1
vote
0
answers
204
views
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 $x \in M$, ...
- 195
1
vote
0
answers
82
views
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 ...
- 111
1
vote
0
answers
121
views
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 ...
- 708
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vote
0
answers
127
views
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 ...
- 11
1
vote
0
answers
73
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 $\alpha^{n-1,n-2}$ ...
- 43
1
vote
0
answers
92
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
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vote
0
answers
77
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 ...
- 2,095