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Keywords:
natural bundles; natural operations
natural bundles; natural operations
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.
References:
[1] Atiyah, M.: Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc. 85 (1957), 181–207. DOI 10.1090/S0002-9947-1957-0086359-5 | MR 0086359
[2] Atiyah, M., Bott, R., Patodi, V.K.: On the heat equation and the index theorem. Invent. Math. 19 (1973), 279–330. DOI 10.1007/BF01425417 | MR 0650828
[3] Bernig, A.: Natural operations on differential forms on contact manifolds. Differential Geom. Appl. 50 (2017), 34–51. DOI 10.1016/j.difgeo.2016.10.005 | MR 3588639
[4] Epstein, D.B.A., Thurston, W.P.: Transformation groups and natural bundles. Proc. Lond. Math. Soc. 38 (1976), 219–236. MR 0531161
[5] Freed, D.S., Hopkins, M.J.: Chern-Weil forms and abstract homotopy theory. Bull. Amer. Math. Soc. 50 (2013), 431–468. DOI 10.1090/S0273-0979-2013-01415-0 | MR 3049871
[6] Goodman, R., Wallach, N.: Representation and Invariants of the Classical Groups. Cambridge University Press, 1998. MR 1606831
[7] Gordillo, A., Navarro, J., Sancho, P.: A remark on the invariant theory of real Lie groups. Colloq. Math., to appear.
[8] Katsylo, P.I., Timashev, D.A.: Natural differential operations on manifolds: an algebraic approach. Sbornik: Mathematics 199 (2008), 1481–1503. DOI 10.1070/SM2008v199n10ABEH003969 | MR 2473812
[9] Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer-Verlag, Berlin, 1993. MR 1202431
[10] Krupka, D., Mikolášová, V.: On the uniqueness of some differential invariants: $\,d, [,], \nabla $. Czechoslovak Math. J. 34 (1984), 588–597. MR 0764440
[11] Mason-Brown, L.: Natural structures in differential geometry. private communication.
[12] Navarro, J., Sancho, J.B.: Peetre-Slovák’s theorem revisited. arXiv: 1411.7499.
[13] Navarro, J., Sancho, J.B.: Natural operations on differential forms. Differential Geom. Appl. 38 (2015), 159–174. DOI 10.1016/j.difgeo.2014.12.003 | MR 3304675
[14] Nijenhuis, A.: Natural bundles and their general properties. Differential Geometry in honor of K. Yano, Kinokuniya, Tokyo, 1972, pp. 317–334. MR 0380862 | Zbl 0246.53018
[15] Palais, R.S.: Natural operations on differential forms. Trans. Amer. Math. Soc. 92 (1959), 125–141. DOI 10.1090/S0002-9947-1959-0116352-7 | MR 0116352
[16] Terng, C.L.: Natural vector bundles and natural differential operators. Amer. J. Math. 100 (1978), 775–828. DOI 10.2307/2373910 | MR 0509074 | Zbl 0422.58001
[17] Timashev, D.A.: On differential characteristic classes of metrics and connections. Fundam. Priklad. Mat. 20 (2015), 167–183. MR 3472276
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