ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53 Deszcz, Ryszard, Agricultural University of Wrocław, Poland On the equivalence of Ricci-semisymmetry and semisymmetry Every semisymmetric (R(X, Y) · R = 0) semi-Riemannian manifold (M, g), dim M ≥ 3, is Ricci-semisymmetric (R(X, Y) · S = 0), where R and S is the curvature tensor and the Ricci tensor of (M, g), respectively, and R(X, Y)· denote derivations induced by R. The converse statement is not true. However, under some additional assumptions both conditions are equivalent for certain manifolds. A review of results related to this problem will be presented. For instance, these conditions are equivalent for non-Einstein Riemannian manifolds with pseudosymmetric Weyl tensor [1] as well as for 4-dimensional hypersurfaces in semi-Riemannian space forms [2]. References: [1] K. Arslan, Y. C .elik, R. Deszcz and R. Ezentas., On the equivalence of Ricci-semisymmetry and semisymmetry, Colloquium Math. 76 (1998), 279–294.

[2] F. Defever, R. Deszcz, Z. S.entürk, L. Verstraelen and S.. Yaprak, P. J. Ryan’s problem in semi-Riemannian space forms, Glasgow Math. J. , in print.