Previous |  Up |  Next


Title: Natural operations on holomorphic forms (English)
Author: Navarro, A.
Author: Navarro, J.
Author: Tejero Prieto, C.
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 54
Issue: 4
Year: 2018
Pages: 239-254
Summary lang: English
Category: math
Summary: We prove that the only natural differential operations between holomorphic forms on a complex manifold are those obtained using linear combinations, the exterior product and the exterior differential. In order to accomplish this task we first develop the basics of the theory of natural holomorphic bundles over a fixed manifold, making explicit its Galoisian structure by proving a categorical equivalence à la Galois. (English)
Keyword: natural bundles
Keyword: natural operations
MSC: 32L05
MSC: 58A32
idZBL: Zbl 06997353
idMR: MR3887363
DOI: 10.5817/AM2018-4-239
Date available: 2018-12-06T16:10:55Z
Last updated: 2020-01-05
Stable URL:
Reference: [1] Atiyah, M.: Complex analytic connections in fibre bundles.Trans. Amer. Math. Soc. 85 (1957), 181–207. MR 0086359, 10.1090/S0002-9947-1957-0086359-5
Reference: [2] Atiyah, M., Bott, R., Patodi, V.K.: On the heat equation and the index theorem.Invent. Math. 19 (1973), 279–330. MR 0650828, 10.1007/BF01425417
Reference: [3] Bernig, A.: Natural operations on differential forms on contact manifolds.Differential Geom. Appl. 50 (2017), 34–51. MR 3588639, 10.1016/j.difgeo.2016.10.005
Reference: [4] Epstein, D.B.A., Thurston, W.P.: Transformation groups and natural bundles.Proc. Lond. Math. Soc. 38 (1976), 219–236. MR 0531161
Reference: [5] Freed, D.S., Hopkins, M.J.: Chern-Weil forms and abstract homotopy theory.Bull. Amer. Math. Soc. 50 (2013), 431–468. MR 3049871, 10.1090/S0273-0979-2013-01415-0
Reference: [6] Goodman, R., Wallach, N.: Representation and Invariants of the Classical Groups.Cambridge University Press, 1998. MR 1606831
Reference: [7] Gordillo, A., Navarro, J., Sancho, P.: A remark on the invariant theory of real Lie groups.Colloq. Math., to appear.
Reference: [8] Katsylo, P.I., Timashev, D.A.: Natural differential operations on manifolds: an algebraic approach.Sbornik: Mathematics 199 (2008), 1481–1503. MR 2473812, 10.1070/SM2008v199n10ABEH003969
Reference: [9] Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry.Springer-Verlag, Berlin, 1993. MR 1202431
Reference: [10] Krupka, D., Mikolášová, V.: On the uniqueness of some differential invariants: $\,d, [,], \nabla $.Czechoslovak Math. J. 34 (1984), 588–597. MR 0764440
Reference: [11] Mason-Brown, L.: Natural structures in differential geometry.private communication.
Reference: [12] Navarro, J., Sancho, J.B.: Peetre-Slovák’s theorem revisited.arXiv: 1411.7499.
Reference: [13] Navarro, J., Sancho, J.B.: Natural operations on differential forms.Differential Geom. Appl. 38 (2015), 159–174. MR 3304675, 10.1016/j.difgeo.2014.12.003
Reference: [14] Nijenhuis, A.: Natural bundles and their general properties.Differential Geometry in honor of K. Yano, Kinokuniya, Tokyo, 1972, pp. 317–334. Zbl 0246.53018, MR 0380862
Reference: [15] Palais, R.S.: Natural operations on differential forms.Trans. Amer. Math. Soc. 92 (1959), 125–141. MR 0116352, 10.1090/S0002-9947-1959-0116352-7
Reference: [16] Terng, C.L.: Natural vector bundles and natural differential operators.Amer. J. Math. 100 (1978), 775–828. Zbl 0422.58001, MR 0509074, 10.2307/2373910
Reference: [17] Timashev, D.A.: On differential characteristic classes of metrics and connections.Fundam. Priklad. Mat. 20 (2015), 167–183. MR 3472276


Files Size Format View
ArchMathRetro_054-2018-4_4.pdf 575.1Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo