Osserman Conjecture in dimension n ? 8, 16

Abstract

Let M n be a Riemannian manifold and R its curvature tensor. For a point p ∈ M n and a unit vector X ∈ T p M n , the Jacobi operator is defined by R X =R(X,·)X. The manifold M n is called pointwise Osserman if, for every p ∈ M n , the spectrum of the Jacobi operator does not depend of the choice of X, and is called globally Osserman if it depends neither of X, nor of p. Osserman conjectured that globally Osserman manifolds are two-point homogeneous. We prove the Osserman Conjecture for n≠8, 16, and its pointwise version for n≠2, 4, 8, 16. Partial result in the case n=16 is also given.