The Kähler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kähler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kähler metrics into Kähler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kähler metrics is shown to be exactly the class of Kähler metrics whose potential function is only a function of the distance from the origin in ℂn. Finally we show that any rotational even dimensional hypersurface carries locally a natural Kähler structure which is of quasi-constant holomorphic sectional curvatures.
@article{bwmeta1.element.doi-10_2478_s11533-008-0004-1,
author = {Georgi Ganchev and Vesselka Mihova},
title = {K\"ahler manifolds of quasi-constant holomorphic sectional curvatures},
journal = {Open Mathematics},
volume = {6},
year = {2008},
pages = {43-75},
zbl = {1140.53010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0004-1}
}
Georgi Ganchev; Vesselka Mihova. Kähler manifolds of quasi-constant holomorphic sectional curvatures. Open Mathematics, Tome 6 (2008) pp. 43-75. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0004-1/
[1] Boju V., Popescu M., Espaces à courbure quasi-constante, J. Differential Geom., 1978, 13, 373–383 (in French) | Zbl 0421.53033
[2] Bryant R., Bochner-Kähler metrics, J. Amer. Math. Soc., 2001, 14, 623–715 http://dx.doi.org/10.1090/S0894-0347-01-00366-6
[3] Bishop R., O’Neil B, Manifolds of negative curvature, Trans. Amer. Math. Soc., 1969, 145, 1–49 http://dx.doi.org/10.2307/1995057 | Zbl 0191.52002
[4] Gray A., Hervella L., The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl., 1980, 123, 35–58 http://dx.doi.org/10.1007/BF01796539 | Zbl 0444.53032
[5] Ganchev G., Mihova V., Riemannian manifolds of quasi-constant sectional curvatures, J. Reine Angew. Math., 2000, 522, 119–141 | Zbl 0952.53017
[6] Ganchev G., Mihova V, Kähler metrics generated by functions of the time-like distance in the flat Kähler-Lorentz space, J. Geom. Phys., 2007, 57, 617–640 http://dx.doi.org/10.1016/j.geomphys.2006.05.004 | Zbl 1106.53015
[7] Ganchev G, Mihova V., Warped product Kähler manifolds and Bochner-Kähler metrics, preprint available at http://arxiv.org/abs/math/0605082 | Zbl 1155.53013
[8] Janssens D., Vanhecke L., Almost contact structures and curvature tensors, Kodai Math. J., 1981, 4, 1–27 http://dx.doi.org/10.2996/kmj/1138036310 | Zbl 0472.53043
[9] Kobayashi S., Nomizu K., Foundations of Differential Geometry, Vol. II, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1969 | Zbl 0175.48504
[10] Tashiro Y., On contact structure of hypersurfaces in complex manifolds I, Tôhoku Math. J. (2), 1963, 15, 62–78 http://dx.doi.org/10.2748/tmj/1178243870
[11] Tashiro Y., On contact structure of hypersurfaces in complex manifolds II, Tôhoku Math. J. (2), 1963, 15, 167–175 http://dx.doi.org/10.2748/tmj/1178243843 | Zbl 0126.38003
[12] Tachibana S., Liu R.C., Notes on Kählerian metrics with vanishing Bochner curvature tensor, Kōdai Math. Sem. Rep., 1970, 22, 313–321 http://dx.doi.org/10.2996/kmj/1138846167 | Zbl 0199.25303
[13] Tricerri F., Vanhecke L., Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc., 1981, 267, 365–397 http://dx.doi.org/10.2307/1998660 | Zbl 0484.53014