Osserman manifolds of dimension 8

For a Riemannian manifold Mⁿ with the curvature tensor R, the Jacobi operator R_X is defined by R_XY=R(X,Y)X. The manifold Mⁿ is called pointwise Osserman if, for every pMⁿ, the eigenvalues of the Jacobi operator R_X do not depend of a unit vector XT_pMⁿ, 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 n8,1614]. Here we prove the Osserman Conjecture and its pointwise version for 8-dimensional manifolds.Mathematics Subject Classification (2000): 53B20