A note on the Osserman conjecture and isotropic covariant derivative of curvature

Abstract

Let M be a Riemannian manifold with the Jacobi operator, which has constant eigenvalues, independent on the unit vector X ∈ TpM and the point p ∈ M. Osserman conjectured that these manifolds are flat or rank-one locally symmetric spaces (∇R = 0). It is known that for a general pseudo-Riemannian manifold, the Osserman-type conjecture is not true and 4-dimensional Kleinian Jordan-Osserman manifolds are curvature homogeneous. We show that the length of the first covariant derivative of the curvature tensor is isotropic, i.e. ||∇R|| = 0. For known examples of 4-dimensional Osserman manifolds of signature (- -++) we check also that || ∇R|| = 0. By the presentation of a class of examples we show that curvature homogeneity and ||∇R|| = 0 do not imply local homogeneity; in contrast to the situation in the Riemannian geometry, where it is unknown if the Osserman condition implies local homogeneity. © 1999 American Mathematical Society.