On compact holomorphically pseudosymmetric Kählerian manifolds

Zbigniew Olszak 1
  • 1 Wrocław University of Technology


For compact Kählerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct examples of non-compact essentially holomorphically pseudosymmetric Kählerian manifolds. These examples show that the compactness assumption cannot be omitted in the above stated theorem. Recently, the first examples of compact, simply connected essentially holomorphically pseudosymmetric Kählerian manifolds are discovered in [4]. In these examples, the structure functions change their signs on the manifold.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Boeckx E., Kowalski O., Vanhecke L., Riemannian manifolds of conullity two, World Scientific Publishing Co., Inc., River Edge, NJ, 1996

  • [2] Deszcz R., On pseudosymmetric spaces, Bull. Soc. Math. Belg. Sér. A, 1992, 44, 1–34

  • [3] Hotlos M., On holomorphically pseudosymmetric Kählerian manifolds, In: Geometry and topology of submanifolds, VII (Leuven, 1994/Brussels, 1994), 139–142, World Sci. Publ., River Edge, NJ, 1995

  • [4] Jelonek W., Compact holomorphically pseudosymmetric Kähler manifolds, preprint available at http://arxiv.org/abs/0902.2535 (2009)

  • [5] Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. II, Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996

  • [6] Lichnerowicz A., Courbure, nombres de Betti et espaces symmetriques, In: Proc. Internat. Congress Math. (Cambridge, 1950), Amer. Math. Soc., Vol. II, 1952, 216–223

  • [7] Mikeš J., Holomorphically projective mappings and their generalizations, J. Math. Sci. (New York), 1998, 89, 1334–1353 http://dx.doi.org/10.1007/BF02414875

  • [8] Mikesh J., Radulovich Z., Haddad M., Geodesic and holomorphically projective mappings of m-pseudo- and m-quasisymmetric Riemannian spaces, Russian Math. (iz. VUZ), 1996, 40, 28–32; translation from Izv. Vyssh. Uchebn. Zaved. Mat., 1996, 10(413), 30–35

  • [9] Mirzoyan V.A., Structure theorems for Kählerian Ric-semisymmetric spaces, Dokl. Nats. Akad. Nauk Armen., 1995, 95, 3–5 (in Russian)

  • [10] Olszak Z., Bochner flat Kählerian manifolds with a certain condition on the Ricci tensor, Simon Stevin, 1989, 63, 295–303

  • [11] Sinjukov N.S., Geodesic mappings of Riemannian spaces, “Nauka”, Moscow, 1979 (in Russian)

  • [12] Szabó Z.I., Structure theorems on Riemannian spaces satisfying R(X, Y) · R = 0. I. The local version, J. Differential Geom., 1982, 17, 531–582

  • [13] Szabó Z.I., Structure theorems on Riemannian spaces satisfying R(X, Y)·R = 0. II. Global versions, Geom. Dedicata, 1985, 19, 65–108 http://dx.doi.org/10.1007/BF00233102

  • [14] Yano K., Bochner S., Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953

  • [15] Yaprak Ş, Pseudosymmetry type curvature conditions on Kähler hypersurfaces, Math. J. Toyama Univ., 1995, 18, 107–136


Journal + Issues