The proportionality principle for Osserman manifolds

Abstract

We give the necessary and sufficient conditions for Jacobi operators that determine an algebraic curvature tensor. This motivates us to introduce the new concept of Jacobi-proportional Riemannian tensors, whose special case is the Rakić duality principle. We prove that all known Osserman tensors (both two-root Osserman and Clifford) are Jacobi-proportional. After the results given by Nikolayevsky, it is known that possible counterexamples of the Osserman conjecture can occur in dimension 16 only, while the reduced Jacobi operator has an eigenvalue of multiplicity 7 or 8. We prove that Jacobi-proportional Osserman tensors that do not satisfy the Osserman conjecture are 2-root with multiplicities 8 and 7, or 3-root with multiplicities 7, 7, and 1.