New examples of Osserman metrics with nondiagonalizable Jacobi operators

Four-dimensional Osserman metrics with nondiagonalizable Jacobi operators

“…[7], [12], [16]). However, very recently the authors have shown the existence of a family of Type II Jordan-Osserman metrics with non-nilpotent Jacobi operators [10]. The purpose of this work is to clarify the situation of Type II Jordan-Osserman metrics by proving the following Main Theorem Let (M, g) be a four-dimensional Type II Jordan-Osserman manifold.…”

Four-dimensional Osserman metrics of neutral signature

“…Chi [1988] showed that any Riemannian Osserman 4-manifold is locally isometric to a 2-point homogeneous space; from later work [Blažić et al 1997;García-Río et al 1997], it follows that any Lorentzian 4-manifold has constant sectional curvature. However the situation is much more complicated in neutral signature (2, 2); there exist many examples of nonsymmetric Osserman pseudo-Riemannian manifolds of neutral signature -see [Díaz-Ramos et al 2006b] and [García-Río et al 1998]. Despite the results of [Alekseevsky et al 1999;Blažić et al 2001;Díaz-Ramos et al 2006a;García-Río and Vázquez-Lorenzo 2001], it is still an open problem to completely describe 4-dimensional Osserman metrics of neutral signature.…”

Symplectic, Hermitian and Kähler Structures on Walker 4-Manifolds

“…where a, b and c are functions of the coordinates (x 1 , x 2 , x 3 , x 4 ). Note that D = span {∂ 1 , ∂ 2 } (∂ i = ∂/∂x i ).…”

“…In §3, we also give the integrability condition of J in (3) as a system of PDE's (Theorem 2), which looks similar to the almost Kähler condition in (4), and obtain the Walker metrics which can be Hermitian with respect to J (Theorem 3).…”