On 4-dimensional neutral Osserman manifolds
by
Zoran Rakic
University of Belgrade, Yugoslavia
Coauthors: Novica Blazic (Faculty of Mathematics, University of Belgrade), Neda Bokan (Faculty of Mathematics, University of Belgrade)

Oral Communication

One of the basic problem in differential geometry and some parts of physics is the reconstruction of the geometric structure of pseudo-Riemannian manifold from the curvature tensor. It is an interesting question up to which level the sectional curvatures of a Riemannian manifold could determine metric. The simplest case is if all sectional curvature are same and then M is a space of the constant sectional curvature and so metric are completely determined.

The next case (in the Riemannian settings), by the simplicity, is when the Jacobi operator, K_X:Y --> R(Y, X)X, has the constant eigenvalues (counting multiplicities) independent on an arbitrary unit vector X and on a point of the manifold. Osserman ([O]) conjectured that such manifolds have to be two-point homogenous, i.e. rank-one locally symmetric or flat. The conjecture was confirmed by Chi ([C]) for dimM =/= 4k, k > 1. This condition is called global Osserman condition and such manifold is called Osserman manifold. The manifolds such that the eigenvalues of Jacobi operator are independent of an arbitrary unit vector X, but vary with the point p of M are called pointwise Osserman manifolds.

We gave the generalization of the Osserman condition in pseudo-Riemannian settings in terms of Jordan form of the Jacobi operator.

Let S⁺(p) and S^-(p) be the sets of all unit spacelike and timelike vectors respectively at a point p in M. We say that (M, g) is Osserman manifold if the Jordan form of K_X is independent of the choice X in S⁺(p) \cup S^-(p) for arbitrary point p in M.

Also, Lorentzian Osserman manifolds have to be of constant sectional curvature (see [BBG]).

We are particularly interested in the neutral signature (- - + +) and Osserman condition.

If M is a manifold with the diagonalizable Jacobi operator K_X or if its characteristic polynomial has no triple root then in [BBR] was proved that M is rank-one symmetric space or flat. But for K_X with triple eigenvalue \alpha and the minimal polynomial of the degree 2 or 3 it is not true (see [BBR], [R], [BCGHV], [GVV]). These are especially type II and type III Osserman manifolds. In order to understand better their rich structure we study geometry of non-symmetric Osserman spaces (types II and III). It is known that Osserman 4-dimensional neutral manifolds have to be self-dual ([ABBR]), what is a generalization of same statement for Riemannian manifolds ([GSV]). Here we show that Osserman manifolds of types II and III admit an integrable, autoparallel isotropic 2-dimensional distribution in T_p(M), p in M. These distributions have been studied also in the framework of the conformal structure CO(2, 2) of 4-dimensional manifold M (see [A], [AK]). We study also some coordinate systems for type II and III Osserman manifolds such that the metric has a simpler form. Let us mention that Walker metric ([Wa]), corresponding to the parallel distributions, belongs to this class and the existence of this type of metric was one of the most important fact in the proof that type II Osserman manifold have to have K_X with all equal eigenvalues.

M\. A\. Akivis On the real theory of four dimensional conformal structures J\. Geom\. Physics 21 1996 55-80

M\. A\. Akivis and V\. V\. Konnov Local aspects in conformal structure theory Uspekhi Mat\. Nauk 48 1993 (1) pages 3-40 (Russian); English transl. in Russian Math. Surveys 48 (1993) 1, 1-35

D\. M\. Alekseevsky, N\. Blazi\' c, N\. Bokan and Z\. Raki\' c Self-dual and pointwise Osserman spaces preprint 1997

N\. Blazi\' c, N\. Bokan and P\. Gilkey A Note on Osserman Lorentzian manifolds Bull\. London Math\. Soc\. 29 1997 227-230

N\. Blazi\' c, N\. Bokan and Z\. Raki\'c Osserman pseudo-Riemannian manifolds of signature (2, 2) preprint 1995, 1997

A\. Bonome, R\. Castro, E\. Garcia-Rio, L\. M\. Hervella, R\. Vásquez-Lorenzo Nonsymmetric Osserman indefinite Kähler manifolds Proc. Amer. Math. Soc. 1998

Q\. S\. Chi A curvature characterization of certain locally rank-one symmetric spaces J\. Diff\. Geom 28 1988 187-202

E\. Garcia-Rio, M\. E\. V\' azquez-Abal and R\. V\' azquez-Lorenzo Nonsymmetric Osserman pseudo -Riemannian manifolds Proc. Amer. Math. Soc. 1997

P\. Gilkey, A\. Swann and L\. Vanhecke Isoparametric geodesic spheres and a conjecture of Osserman concerning the Jacobi operator Quart\. J\. Math\. Oxford\. 46 1995 299-320

R\. Osserman Curvature in the eighties Amer\. Math\. Monthly 97 1990 731-756

Z. Raki\' c Rank 2 symmetric Osserman spaces Bull. Austral. Math. Soc. 56 1998 517-521

A\. G\. Walker Canonical form for a Riemannian space with a parallel field of null planes Quart\. J\. Math\. (Oxford) (2) 1 1950 69-79

Date received: May 31, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-04.