  By considering the projectivized spectrum of the Jacobi operator, we
introduce the concept of projective Osserman manifold in both the affine and in
the pseudo-Riemannian settings. If M is an affine projective Osserman manifold,
then the modified Riemannian extension metric on the cotangent bundle is both
spacelike and timelike projective Osserman. Since any rank 1 symmetric space is
affine projective Osserman, this provides additional information concerning the
cotangent bundle of a rank 1 Riemannian symmetric space with the modified
Riemannian extension metric. We construct other examples of affine projective
Osserman manifolds where the Ricci tensor is not symmetric and thus the
connection is not the Levi-Civita connection of any metric. If M is an affine
projective Osserman manifold of odd dimension, we use methods of algebraic
topology to show the Jacobi operator has only one non-zero eigenvalue and that
eigenvalue is real.
