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 $X\in T_{p}M$ and the point $p\in M$. Osserman conjectured that these manifolds are flat or rank-one locally symmetric spaces ($\nabla R =0$). 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. $\Vert \nabla R\Vert =0$. For known examples of 4-dimensional Osserman manifolds of signature $(–++)$ we check also that $\Vert \nabla R\Vert =0$. By the presentation of a class of examples we show that curvature homogeneity and $\Vert \nabla R\Vert =0$ 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.