All Questions
Tagged with differential-forms dg.differential-geometry
58 questions
5
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0
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98
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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 ...
- 51
3
votes
0
answers
102
views
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 ...
- 161
4
votes
1
answer
179
views
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 ...
- 41
1
vote
0
answers
204
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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 $x \in M$, ...
- 195
1
vote
0
answers
82
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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 ...
- 111
0
votes
0
answers
84
views
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
$$f(u, v)=(a\sin(u) \cos(v), b \...
- 251
15
votes
2
answers
889
views
Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?
Fix a compact Riemannian manifold $M$ (leaving the metric implicit). What I'd like to know is if the corresponding Hodge decomposition of smooth $n$-forms
$$
\Omega^n(M) \simeq \mathcal{H}^n(M)\oplus ...
- 35.5k
6
votes
1
answer
404
views
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 ...
- 833
2
votes
0
answers
65
views
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 $\bar{\partial}_V: \Omega^{p,q}(V) \rightarrow \Omega^{p,...
- 121
2
votes
1
answer
199
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Decomposition of forms on a Spin$(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
0
votes
1
answer
193
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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 ...
- 103
4
votes
1
answer
248
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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 ...
- 305
2
votes
0
answers
135
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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....
2
votes
0
answers
135
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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
3
votes
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
9
votes
1
answer
868
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 ...
- 1,545
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
10
votes
2
answers
892
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
2k
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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. ...
2
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 $\...
- 4,195
2
votes
1
answer
193
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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{...
- 139
13
votes
2
answers
1k
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
1
vote
0
answers
328
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 $\Lambda^2(M)\to\Lambda^1(M)$. ...
- 213
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 {...
- 41
5
votes
1
answer
147
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$ ...
- 15.5k
8
votes
1
answer
484
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 ...
- 35.5k
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 ...
10
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2
answers
1k
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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 ...
- 502
3
votes
1
answer
390
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 ...
- 449
7
votes
0
answers
282
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\...
- 366
5
votes
1
answer
637
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 ...
- 366
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
1
answer
201
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}^...
- 993
3
votes
0
answers
298
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 ...
- 366
7
votes
7
answers
503
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 differential topology ...
- 261
8
votes
1
answer
329
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 $\beta$...
- 797
1
vote
1
answer
239
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
11
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0
answers
354
views
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, ...
- 1,611
6
votes
1
answer
163
views
Restriction of "Spin(7) 4-form" to $\mathbb{R}_+\times S^7$
I asked this question on stackexchange (https://math.stackexchange.com/questions/2212226/restriction-of-spin7-4-form-to-mathbbr-times-s7) but was advised to ask again here:
I'm currently reading ...
8
votes
1
answer
2k
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What do the differential k-forms on a product manifold look like?
I am interested in how I could express $\Omega^k( M \times N)$ in terms of $\Omega^i(M)$ and $\Omega^j(N)$ for $i,j = 0,1, \ldots k$. Is there a nice relation?
This question arose in the context of ...
- 313
3
votes
0
answers
1k
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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 ...
- 327
0
votes
1
answer
604
views
Exterior derivative on principal bundle [closed]
In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...
- 103
5
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0
answers
82
views
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 ...
- 191
4
votes
1
answer
455
views
Closed $3$-manifold, $2$-dimensional subbundle of this manifold, is this form exact or not?
Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. I know the following.
There is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector ...
user74565
13
votes
3
answers
787
views
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 ...
- 735
14
votes
0
answers
574
views
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 ...
- 37.8k
6
votes
1
answer
291
views
Strange problem about triplets of differential forms
Suppose we have the following map:
$$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$
$$(\alpha,\beta,\gamma)\longmapsto(\mathrm{d}\alpha+\beta\wedge\gamma,\mathrm{d}\beta+\...
- 2,091
3
votes
0
answers
304
views
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 ...
- 203
5
votes
3
answers
1k
views
Non-continuous differentiability for differential forms
Generally when working with differential forms, one assumes that they are continuously differentiable, i.e. $C^r$ for some $1\le r \le \infty$. Under this hypothesis, one can define the exterior ...
- 66.8k
2
votes
2
answers
657
views
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 ...
- 21