Complex and symplectic geometry of vector bundle manifolds

M. T. K. Abbassi ; R. El Masdouri ; I. Lakrini

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1295-1312.

Résumé

The aim of this paper is to explore the complex and symplectic geometries of vector bundle manifolds. We will construct an almost complex structure on total spaces of vector bundles, endowed with a complex structure, over an almost complex base. Then we give necessary and sufficient conditions for its integrability. Meanwhile, we accomplish a symplectic version of this construction. We construct almost symplectic structures on vector bundle manifolds and we characterize those which are symplectic on the total space. Finally, we apply the constructions to the case of tangent bundles and Whitney sums. In particular, we obtain an infinite family of non-compact flat Kähler manifolds.

Détail

Keywords: (almost) complex structure, symplectic structure, Kähler manifold, vector bundle, spherically symmetric metric.

Bibliographie

[1] M.T.K. Abbassi, “$g$-natural metrics: new horizons in the geometry of tangent bundles of Riemannian manifolds”, Note. Mat., 28, suppl. 1 (2009), 6–35 | MR | Zbl

[2] M.T.K. Abbassi, Métriques Naturelles Riemanniennes sur le Fibrùé tangent à une variété Riemannienne, Editions Universitaires Européénnes, Saarbrücken, Germany, 2012

[3] M.T.K. Abbassi, M. Sarih, “On some hereditary properties of Riemannian $g$-natural metrics on tangent bundles of Riemannian manifolds”, Differ. Geom. Appl., 22:1 (2005), 19–47 | DOI | MR | Zbl

[4] M.T.K Abbassi, I. Lakrini, “On the geometry of vector bundles with flat connections”, Bull. Korean Math. Soc., 56:5 (2019), 1219–1233 | MR | Zbl

[5] M.T.K Abbassi, I. Lakrini, “On the completeness of total space of horizontally conformal submersions”, Commun. Math., 29:3 (2021), 493–504 | DOI | MR | Zbl

[6] M.T.K Abbassi, I. Lakrini, “Conformal vector fields on vector bundle manifolds with spherically symmetric metrics”, Publ. Math. Debr., 97:3-4 (2020), 289–311 | DOI | MR | Zbl

[7] R.M. Aguilar, “Isotropic almost complex structures on tangent bundles”, Manuscr. Math., 90:4 (1996), 429–436 | DOI | MR | Zbl

[8] R. Albuquerque, “On vector bundle manifolds with spherically symmetric metrics”, Ann. Global Anal. Geom., 51:2 (2017), 129–154 | DOI | MR | Zbl

[9] V.I. Arnold, Mathematical methods of classical mechanics, Springer, New York etc, 1978 | MR | Zbl

[10] L. Auslander, M. Kuranishi, “On the holonomy group of locally Euclidean spaces”, Ann. Math. (2), 65:3 (1957), 441–415 | DOI | MR | Zbl

[11] A. Baghban, E. Abedi, “Isotropic almost complex structures and harmonic unit vector fields”, Arch. Math. Brno, 54:1 (2018), 15–32 | DOI | MR | Zbl

[12] M. Benyounes, E. Loubeau, C.M. Wood, “Harmonic sections of Riemannian vector bundles, and metrics of Cheeger-Gromoll type”, Differ. Geom. Appl., 25:3 (2007), 322–334 | DOI | MR | Zbl

[13] L.S. Charlap, “Compact flat Riemannian manifolds I”, Ann. Math. (2), 81:1 (1965), 15–30 | DOI | MR | Zbl

[14] L.S. Charlap, A.T. Vasquez, “Compact flat Riemannian manifolds II: The cohomology of $Z_p$-manifolds”, Am. J. Math., 87:3 (1965), 551–563 | DOI | MR | Zbl

[15] L.S. Charlap, A.T. Alphonse, “Compact flat Riemannian manifolds. III: The group of affinities”, Am. J. Math., 95:3 (1973), 471–494 | DOI | MR | Zbl

[16] L.S. Charlap, Bieberbach groups and flat manifolds, Springer-Verlag, New York etc, 1986 | MR | Zbl

[17] K. Dekimpe, M. Hałenda, A. Szczepański, “Kähler flat manifolds”, J. Math. Soc. Japan, 61:2 (2009), 363–377 | DOI | MR | Zbl

[18] P. Dombrowski, “On the geometry of the tangent bundles”, J. Reine Angew. Math., 210 (1962), 73–88 | DOI | MR | Zbl

[19] S. Dragomir, D. Perrone, Harmonic vector fields. Variational principles and differential geometry, Elsevier, Amsterdam, 2012 | MR | Zbl

[20] S.I. Goldberg, Curvature and homology, Academic Press, New York, London, 1962 | MR | Zbl

[21] F.E.A. Johnson, E.G. Rees, “Kähler groups and rigidity phenomena”, Math. Proc. Camb. Philos. Soc., 109:1 (1991), 31–44 | DOI | MR | Zbl

[22] J.J. Konderak, “On sections of fibre bundles which are harmonic maps”, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér., 42(90):4 (1999), 341–352 | MR

[23] O. Kowalski, “Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold”, J. Reine Angew. Math., 250 (1971), 124–129 | MR | Zbl

[24] O. Kowalski, M. Sekizawa, “Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles – a classification”, Bull. Tokyo Gakugei Univ., Sect. IV, Ser. Math. Nat. Sci., 40 (1988), 1–29 | MR | Zbl

[25] I. Kolář , P.W. Michor, J. Slovák, Natural operations in differential geometry, Springer-Verlag, Berlin, 1993 | MR | Zbl

[26] S. Kobayashi, K. Nomizu, Foundations of differential geometry, v. II, Interscince Publishers, New York-London-Sydney, 1963 | MR | Zbl

[27] A. Lichnerowicz, “Généralizations de la géométrie Kählerienne globale”, Colloque de Géométrie Différentielle (Louvin, 1951) | MR

[28] K-P. Mok, “Metrics and connections on the cotangent bundle”, Kōdai Math. Semin. Rep., 28 (1977), 226–238 | MR | Zbl

[29] E. Musso, F. Tricerri, “Riemannian metrics on tangent bundles”, Ann. Mat. Pura Appl., IV. Ser., 150:4 (1988), 1–19 | MR | Zbl

[30] M.I. Munteanu, “Old and new structures on the tangent bundle”, Proceedings of the 8th international conference on geometry, integrability and quantization (Sts. Constantine and Elena, Bulgaria, June 9-14, 2006), eds. Mladenov Ivaïlo et al., Bulgarian Academy of Sciences, Sofia, 2007, 264–278 | MR | Zbl

[31] V. Oproiu, “A generalization of natural almost Hermitian structures on the tangent bundle”, Math. J. Toyama Univ., 22 (1999), 1–14 | MR | Zbl

[32] W.A. Poor, Differential geometric structures, Dover Pubs. Inc., Mineola, New York, 2007; 1981 | Zbl

[33] A.A. Salimov, F. Agca, “Some properties of Sasakian metrics in cotangent bundles”, Mediterr. J. Math., 8:2 (2011), 243–255 | DOI | MR | Zbl

[34] S. Sasaki, “On the differential geometry of tangent bundles of Riemannian manifolds”, Tôhoku Math. J., II. Ser., 10 (1958), 338–354 | MR | Zbl

[35] M. Sekizawa, “Natural transformations of affine connections on manifolds to metrics on cotangent bundles”, Abstract analysis, Proc. 14th Winter Sch. (Srno/Czech, 1986), Suppl. Rend. Circ. Mat. Palermo, II. Ser., 14, 1987, 129–142 | MR | Zbl

[36] M. Tahara, L. Vanhecke, Y. Watanabe, “New structures on tangent bundles”, Note Mat., 18:1 (1998), 131–141 | MR | Zbl

[37] K. Yano, S. Ishihara, Tangent and cotangent bundles. Differential geometry, Marcel Dekker Inc., New York, 1973 | MR | Zbl

[38] S.-T. Yau, “Compact flat Riemannian manifolds”, J. Differ. Geom., 6 (1972), 395–402 | DOI | MR | Zbl