Two theorems on Osserman manifolds

Abstract

Let M n be a Riemannian manifold. 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 , where R is the curvature tensor. 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. R. Osserman conjectured that globally Osserman manifolds are two-point homogeneous. We prove the following: (1) A pointwise Osserman manifold M n is two-point homogeneous, provided 8∤n and n≠2, 4; a globally Osserman manifold M n is two-point homogeneous, provided 8∤n; (2) Let M n be a globally Osserman manifold with the Jacobi operator having exactly two eigenvalues. In the case n=16, assume that the multiplicities of the eigenvalues are not 7 and 8, respectively. Then M n is two-point homogeneous.