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- {{Short description|Conformal structure admits a Hodge dual of 1-forms without even specifying a metric}} ...{harvtxt|Hodge|1941}}. This article covers general results on differential forms on a Riemann surface that do not rely on any choice of [[Riemannian manifol ...78 KB (12,391 words) - 23:20, 25 March 2024
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- ...=Problems And Solutions In Differential Geometry, Lie Series, Differential Forms, Relativity And Applications|first=Willi-hans|last=Steeb|publisher=World Sc ...832 bytes (105 words) - 01:49, 28 April 2024
- ...e notion of the [[integral]] to [[manifold]]s. Rather than functions or [[differential form]]s, the integral is defined over [[current (mathematics)|currents]] on ...s there is a pairing between {{mvar|k}}-currents {{mvar|T}} and {{mvar|k}}-forms {{mvar|α}}, denoted here by ...2 KB (298 words) - 07:32, 6 April 2023
- In differential geometry, an '''equivariant differential form''' on a manifold ''M'' [[Lie group action|acted upon]] by a [[Lie grou ...eratorname{Lie}(G)</math> to the space of [[Differential form|differential forms]] on ''M'' that are equivariant; i.e., ...3 KB (408 words) - 00:09, 23 October 2022
- In [[differential geometry]], a field in [[mathematics]], a '''multivector field''', '''polyv == Comparison with differential forms == ...4 KB (676 words) - 18:43, 18 August 2024
- ...differential forms''' on the standard [[Simplex|''n''-simplex]] is the [[differential graded algebra]]:<ref>{{harvnb|Hinich|1997|loc=§ 4.8.1.}}</ref> * https://ncatlab.org/nlab/show/differential+forms+on+simplices ...1 KB (165 words) - 06:23, 13 May 2024
- ...The covariant derivative defines a degree-1 [[endomorphism]] ''d'', the '''differential''' of <math>\Omega_X^*(E)</math>, and the flatness condition is equivalent ...bundle|canonical line bundle]] <math>\Lambda^{\mathrm{top}}M</math> of a [[differential manifold]] ''M'' is a flat line bundle, called the '''orientation bundle''' ...3 KB (420 words) - 23:26, 21 September 2021
- In [[differential geometry]], the '''tensor product''' of [[vector bundle]]s {{mvar|E}}, {{mv ...omorphism classes of all line bundles on some topological space {{mvar|X}} forms an abelian group called the [[Picard group]] of {{mvar|X}}. ...2 KB (360 words) - 22:10, 13 February 2025
- ...h>\Omega^k_M</math> be the sheaf of [[differential form|differential ''k''-forms]] on a manifold ''M''. If ''n'' is the dimension of ''M'', then the sheaf ...is is that, while one can [[integration of a differential form|integrate a differential form]] only if the manifold is [[oriented manifold|oriented]], one can alwa ...2 KB (292 words) - 20:32, 1 October 2024
- ...ere exists an exterior derivative. Correspondingly, a first order quantum differential calculus means at least the following: ...of <math>\Omega^1</math> to include analogues of higher order differential forms ...6 KB (929 words) - 15:20, 18 May 2023
- Poincaré duality for differential forms is then the conjunction, for ''M'' connected, of the following two statemen * The cup product, or [[exterior product]] of differential forms ...3 KB (395 words) - 17:42, 9 April 2021
- {{short description|Group of differential equations}} ...ast=van Kampen |first=E. R. |date=1941 |title=Notes on Systems of Ordinary Differential Equations |url=https://www.jstor.org/stable/2371531 |journal=American Journ ...7 KB (1,036 words) - 16:22, 3 February 2025
- ==Statement in classical differential geometry== * {{citation|last1 = Struik|first1=Dirk J.|title=Lectures on Classical Differential Geometry|year = 1988|publisher = Dover Publications|page=108|isbn=0-486-656 ...2 KB (348 words) - 03:20, 8 February 2024
- Given a graded manifold <math>(Z,A)</math>, [[differential algebra|graded derivations]] of the structure ring of graded functions <mat ==Graded exterior forms== ...7 KB (1,200 words) - 12:42, 12 June 2022
- ...tial algebraic equation (PDAE)''' set is an incomplete system of [[partial differential equation]]s that is closed with a set of [[algebraic equation]]s. ...e relationship between an [[ordinary differential equation]] (ODE) and a [[differential algebraic equation]] (DAE). ...3 KB (402 words) - 02:07, 7 December 2024
- ...= dx^j - i dy^j</math> on <math>U</math>. Dual to these complex-valued one-forms are the complex-valued vector fields (that is, sections of the complexified ...hic tangent bundles previously described. Similarly the complex-valued one-forms <math>dz^j</math> and <math>d\bar{z}^j</math> provide the splitting of the ...8 KB (1,254 words) - 22:11, 4 March 2024
- ...r form]]s is another modular form, generalizing the product of two modular forms. ...ions for [[polynomial]]s in [[derivative]]s of modular forms to be modular forms, and {{harvs|txt|last=Cohen|authorlink=Henri Cohen (number theorist)|year=1 ...3 KB (520 words) - 19:55, 4 June 2024
- ...>-module''' or '''DG-module''' is a graded module <math>M</math> with a '''differential''' <math>d \colon M \to M \colon M_i \to M_{i+1}</math> making <math>M</mat ...GAlgebra''' is a graded algebra that is a differential graded module whose differential obeys the graded Leibniz rule. ...7 KB (1,141 words) - 22:22, 19 July 2024
- ...ic differential operator]] defined on a certain subspace of the space of [[differential form]]s on an even-dimensional [[closed manifold|compact]] [[Riemannian man be the [[exterior derivative]] on <math>i</math>-th order [[differential forms]] on <math>M</math>. The Riemannian metric on <math>M</math> allows us to d ...5 KB (751 words) - 14:27, 23 October 2022
- This product can be understood as induced by the exterior product of [[differential form]]s in [[de Rham cohomology]]. It makes the singular cohomology of a co *[[Differential graded algebra]]: the algebraic structure arising on the cochain level for ...1 KB (208 words) - 15:36, 8 January 2025
- ...e '''residue formula''' says that the sum of the residues of a meromorphic differential form on a [[smooth morphism|smooth]] [[proper morphism|proper]] [[algebraic ...athematics)|field]] ''k''. A meromorphic [[Kähler differential|(algebraic) differential form]] <math>\omega</math> has, at each [[closed point]] ''x'' in ''X'', a ...2 KB (353 words) - 02:56, 19 February 2025