  Let M be a Riemannian manifold and R its curvature tensor. For a unit vector
X tangent to M at a point p, the Jacobi operator is defined by R_X = R(X, .)
X$. The manifold M is called pointwise Osserman if, for every point p, 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 Osserman Conjecture for dimension n \ne 8, 16, and its pointwise
version for n \ne 2,4,8,16. Partial results in the cases n = 8, 16 are also
given.

Differential Geometry