6 research outputs found
Higher order Jordan Osserman Pseudo-Riemannian manifolds
We study the higher order Jacobi operator in pseudo-Riemannian geometry. We
exhibit a family of manifolds so that this operator has constant Jordan normal
form on the Grassmannian of subspaces of signature (r,s) for certain values of
(r,s). These pseudo-Riemannian manifolds are new and non-trivial examples of
higher order Osserman manifolds
Curvature homogeneous spacelike Jordan Osserman pseudo-Riemannian manifolds
Let s be at least 2. We construct Ricci flat pseudo-Riemannian manifolds of
signature (2s,s) which are not locally homogeneous but whose curvature tensors
never the less exhibit a number of important symmetry properties. They are
curvature homogeneous; their curvature tensor is modeled on that of a local
symmetric space. They are spacelike Jordan Osserman with a Jacobi operator
which is nilpotent of order 3; they are not timelike Jordan Osserman. They are
k-spacelike higher order Jordan Osserman for ; they are k-timelike
higher order Jordan Osserman for , and they are not k timelike
higher order Jordan Osserman for .Comment: Update bibliography, fix minor misprint
Examples of signature (2,2) manifolds with commuting curvature operators
We exhibit Walker manifolds of signature (2,2) with various commutativity
properties for the Ricci operator, the skew-symmetric curvature operator, and
the Jacobi operator. If the Walker metric is a Riemannian extension of an
underlying affine structure A, these properties are related to the Ricci tensor
of A