Einstein-like manifolds which are not Einstein
by
Jürgen Berndt
University of Hull, England

Oral Communication

In 1978 Alfred Gray suggested in [2] several generalizations of Einstein manifolds. His idea was as follows. Any Einstein manifold has parallel Ricci tensor, and conversely, any Riemannian manifold with parallel Ricci tensor is locally a Riemannian product of Einstein manifolds. Thus the Einstein condition is essentially equivalent to the parallelity of the Ricci tensor. Now consider the covariant derivative ÑS of the Ricci tensor S of a Riemannian manifold M as a tensor field of type (0, 3). The algebraic curvature identities and the second Bianchi identity of the Riemannian curvature tensor of M imply certain algebraic identities for ÑS. Let R be the real vector bundle over M whose fibre R_p at p consists of all trilinear maps on T_pM satisfying these algebraic identities. The orthogonal group O(n), n = dimM, acts in a standard way on R, turning each fibre R_p into an O(n)-module. If n > 2 then R_p decomposes into three irreducible O(n)-modules, inducing a direct sum decomposition R = B\oplusA\oplusQ. This decomposition provides naturally several types of generalizations of the Einstein condition. A detailed discussion of this can be found in Chapter 16 of [1].

The bundle B. The covariant derivative ÑS of the Ricci tensor S of a Riemannian manifold M is a section in B if and only if M has harmonic curvature, that is, if the divergence of the Riemannian curvature tensor of M vanishes. Compact manifolds with harmonic curvature are precisely those manifolds for which the Levi Civita connection is a critical point of the Yang-Mills functional \int_M |R^Ñ|² on the tangent bundle TM.

The bundle A. The covariant derivative ÑS of the Ricci tensor S of a Riemannian manifold M is a section in A if and only if S is a Killing tensor, that is, if S is a first integral of the geodesic Hamilton-Jacobi equation on M.

The bundle Q. The covariant derivative ÑS of the Ricci tensor S of a Riemannian manifold M is a section in Q if and only if

Ñ æ è S -  1 2n-2 sg ö ø =  n-2 2(n+2)(n-1) ds \odot g ,

where n is the dimension, s the scalar curvature and g the Riemannian metric of M, and ds \odot g denotes the symmetric product of ds and g.

Many examples are known of manifolds belonging to the classes A and B, but the PDE system characterizing class Q appears to be rather awkward and only very little is known about it. The purpose of the talk is to present a local classification of all 3-dimensional Riemannian metrics solving this PDE system. The method is to relate this problem to the classical theory about complete integrability of the geodesic Hamilton-Jacobi equation by separation of variables.

References

1.

Besse, A.L., Einstein manifolds (Springer-Verlag, Berlin Heidelberg, 1987).

2.

Gray, A., Einstein-like manifolds which are not Einstein, Geom. Dedicata 7 (1978), 259-280.

Homepage of Jurgen Berndt

Date received: May 8, 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 # cadq-40.