Osserman manifolds of dimension 8

Abstract

For a Riemannian manifold Mn with the curvature tensor R, the Jacobi operator RX is defined by RXY=R(X,Y)X. The manifold Mn is called pointwise Osserman if, for every p ∈ Mn, the eigenvalues of the Jacobi operator RX do not depend of a unit vector X ∈ TpMn, and is called globally Osserman if they do not depend of the point p either. R. Osserman conjectured that globally Osserman manifolds are flat or locally rank-one symmetric. This Conjecture is true for manifolds of dimension n≠8,16[14]. Here we prove the Osserman Conjecture and its pointwise version for 8-dimensional manifolds.