On compact holomorphically pseudosymmetric Kählerian manifolds

Zbigniew Olszak

Open Mathematics (2009)

  • Volume: 7, Issue: 3, page 442-451
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Zbigniew Olszak. "On compact holomorphically pseudosymmetric Kählerian manifolds." Open Mathematics 7.3 (2009): 442-451. <http://eudml.org/doc/269598>.

@article{ZbigniewOlszak2009,
abstract = {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.},
author = {Zbigniew Olszak},
journal = {Open Mathematics},
keywords = {Kählerian manifold; Semisymmetry; Holomorphic pseudosymmetry; Kähler manifold; holomorphic pseudosymmetry; holomorphic Ricci-pseudosymmetry; semisymmetry; Ricci-symmetry},
language = {eng},
number = {3},
pages = {442-451},
title = {On compact holomorphically pseudosymmetric Kählerian manifolds},
url = {http://eudml.org/doc/269598},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Zbigniew Olszak
TI - On compact holomorphically pseudosymmetric Kählerian manifolds
JO - Open Mathematics
PY - 2009
VL - 7
IS - 3
SP - 442
EP - 451
AB - 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.
LA - eng
KW - Kählerian manifold; Semisymmetry; Holomorphic pseudosymmetry; Kähler manifold; holomorphic pseudosymmetry; holomorphic Ricci-pseudosymmetry; semisymmetry; Ricci-symmetry
UR - http://eudml.org/doc/269598
ER -

References

top
  1. [1] Boeckx E., Kowalski O., Vanhecke L., Riemannian manifolds of conullity two, World Scientific Publishing Co., Inc., River Edge, NJ, 1996 Zbl0904.53006
  2. [2] Deszcz R., On pseudosymmetric spaces, Bull. Soc. Math. Belg. Sér. A, 1992, 44, 1–34 Zbl0808.53012
  3. [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. [4] Jelonek W., Compact holomorphically pseudosymmetric Kähler manifolds, preprint available at http://arxiv.org/abs/0902.2535 (2009) Zbl1185.53080
  5. [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. [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. [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 Zbl0983.53013
  8. [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. [9] Mirzoyan V.A., Structure theorems for Kählerian Ric-semisymmetric spaces, Dokl. Nats. Akad. Nauk Armen., 1995, 95, 3–5 (in Russian) 
  10. [10] Olszak Z., Bochner flat Kählerian manifolds with a certain condition on the Ricci tensor, Simon Stevin, 1989, 63, 295–303 Zbl0629.53059
  11. [11] Sinjukov N.S., Geodesic mappings of Riemannian spaces, “Nauka”, Moscow, 1979 (in Russian) 
  12. [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 Zbl0508.53025
  13. [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 Zbl0612.53023
  14. [14] Yano K., Bochner S., Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953 
  15. [15] Yaprak Ş, Pseudosymmetry type curvature conditions on Kähler hypersurfaces, Math. J. Toyama Univ., 1995, 18, 107–136 Zbl0849.53039

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.