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...l form]]s, the integral is defined over [[current (mathematics)|currents]] on a manifold. ...stributions]], of the space of {{mvar|k}}-forms {{math|Ω^''k''}} on {{mvar|M}}. Thus there is a pairing between {{mvar|k}}-currents {{mvar|T}} ...

...> of dimension ''n'' is a [[locally constant sheaf]] ''o''_''X'' on ''X'' such that the stalk of ''o''_''X'' at a point ''x'' is the ...^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 ...

...h>H</math>. A Hamiltonian diffeomorphism <math>\phi: M \to M</math> is the integration of a 1-parameter family of Hamiltonian vector fields <math>V_t, t \in [0, 1 ...ürel2015}} On the other hand, it is proved for various types of symplectic manifolds. ...

{{Short description|Extends Poincaré duality beyond oriented manifolds}} ..., the '''twisted Poincaré duality''' is a theorem removing the restriction on [[Poincaré duality]] to [[oriented manifold]]s. The existence of a global o ...

...fold. Their further research dealt with [[conjugate point]]s on Riemannian manifolds. ...re and an application to the equation <math>\Delta u = 1 - Ke^{\ u}</math> on <math>S^2</math>|year=1972|last1=Kazdan|first1=Jerry L.|last2=Warner|first2 ...

...the fiber, via "[[Integral|integration]]". It is also called the '''fiber integration'''. ...old]] with compact oriented fibers. If <math>\alpha</math> is a ''k''-form on ''E'', then for tangent vectors ''w''_''i'''s at ''b'', let ...

...geometry]], '''Santaló's formula''' describes how to integrate a function on the unit [[sphere bundle]] of a [[Riemannian manifold]] by first integratin ...sults.<ref>Ilmavirta, Joonas, and François Monard. "4 Integral geometry on manifolds with boundary and applications." The Radon Transform: The First 100 Years a ...

...of [[topological space]]s and a ring ''R'', the '''pullback''' along ''f'' on [[cohomology theory]] is a [[graded ring|grade]]-preserving ''R''-[[algebra In contrast, a pushforward for de Rham cohomology for example is given by [[integration-along-fibers]]. ...

...[[de Rham cohomology]] to the [[singular cohomology]] given by [[integral|integration]] is an [[isomorphism]]. ...dix D. to {{cite book |last1=Conlon |first1=Lawrence |title=Differentiable Manifolds |date=2001 |publisher=Springer |edition=2nd |doi=10.1007/978-0-8176-4767-4 ...

...s to the attempt in 1895 to define [[homology theory]] using only (smooth) manifolds.<ref name="Hist">{{cite book |last=Dieudonné |first=Jean |authorlink=Jean D ...ast1=Harman |first1=Peter Michael |title=Wranglers and Physicists: Studies on Cambridge Physics in the Nineteenth Century |date=1985 |publisher=[[Manches ...

...cs, a '''superintegrable Hamiltonian system''' is a [[Hamiltonian system]] on a <math>2n</math>-dimensional [[symplectic manifold]] for which the follow (ii) There exist smooth real functions <math>s_{ij}</math> on <math>N</math> such that the [[Poisson manifold|Poisson bracket]] of integr ...

...odge theory, important in proving fundamental analytical results on Kähler manifolds, such as the [[Ddbar lemma|<math>\partial \bar \partial</math>-lemma]], the ...math>\Lambda</math> was introduced by [[André Weil]] in the first textbook on Kähler geometry, ''Introduction à L’Étude des Variétés Kähleriennes.''<ref> ...

...] <math>\nabla _{\alpha }</math> on [[Differentiable manifold|differential manifolds]] in that it produces a [[tensor]] when applied to a tensor. ...uct the other by [[Derivative|differentiation]] or [[Initial value problem|integration]]. ...

...|doi=10.1007/BF01389853}}</ref> to classify the homotopy types of [[Kähler manifolds]]. ...to H^*(\mathcal{A})</math> on cohomology. Note that the cohomology depends on the field <math>k</math>, over which <math>\mathcal{A}</math> is defined. ...

Gross serves on the editorial boards of the ''[[Journal of Functional Analysis]]'',<ref>{{C ...o [[Yang–Mills theory]] and related mathematical theories such as analysis on [[loop group]]s. ...

...also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces. ...lso a local model of '''calculus on manifolds''', a theory of functions on manifolds. ...

...vative can be found by [[symbolic integration]], thus avoiding [[numerical integration]]. ...o compute area via [[infinitesimals]], an operation that we would now call integration. The origins of differentiation likewise predate the fundamental theorem of ...

| title_orig = ''Traité d'analyse'' (''Treatise on Analysis'') ...alysis]], with the working title '''''Traité d'analyse''''' ('''''Treatise on Analysis'''''). While planning the structure of the work they became more ...

...[https://arxiv.org/abs/q-alg/9709040 ''Deformation Quantization of Poisson Manifolds''], ''Letters of Mathematical Physics'' '''66''', pp. 157&ndash;216.</ref>< ...]] quantization is an '''associative''' unital product <math>\star</math> on the algebra of formal [[power series]] in {{math|''ħ'', ''A''<nowiki>[[</no ...

...2/9789812798411_0027|chapter=Hermitian-Yang-Mills Connection on Non-Kähler Manifolds |title=Mathematical Aspects of String Theory |year=1987 |last1=Li |first1=J ...–Donaldson conjecture]] about the existence of [[Kähler–Einstein metric]]s on [[Fano varieties]], and the [[Thomas–Yau conjecture]] about existence of sp ...