  One says that a smooth manifold M is a pseudo-Riemannian manifold of
signature (p,q) if the tangent bundle TM is equipped with a smooth
non-degenerate symmetric inner product g of signature (p,q). Similarly one says
that M is an affine manifold if TM is equipped with a torsion free connection.
One says g is Osserman if the eigenvalues of the Jacobi operator are constant
on the pseudo-sphere bundles of unit timelike and spacelike vectors. We extend
this concept from the pseudo-Riemannian to the affine setting to define the
notion of a projectively Osserman manifold. This notion is the focus of the
paper. We establish some basic results concerning projectively Osserman
manifolds and exhibit examples of this structure which arise in several
different geometrical contexts.
