A Note on Osserman Lorentzian Manifolds

Abstract

Let p be a point of a Lorentzian manifold M. We show that if M is spacelike Osserman at p, then M has constant sectional curvature at p ; similarly, if M is timelike Osserman at p, then M has constant sectional curvature at p. The reverse implications are immediate. The timelike case and 4-dimensional spacelike case were first studied in [3] ; we use a different approach to this case.

Four-dimensional Osserman metrics with nondiagonalizable Jacobi operators

“…It has been shown in [8] and [11] (see also [2]) that any four-dimensional Osserman metric is locally isometric to a two-point homogeneous space if the signature is either Riemannian or Lorentzian. However the situation is much more complicated for neutral metrics and there exist many examples of nonsymmetric Osserman pseudo-Riemannian manifolds of neutral signature [13].…”

The Structure of Algebraic Covariant Derivative Curvature Tensors

“…In the Lorentzian setting (p = 1), an Osserman manifold has constant sectional curvature [2,8]. In the higher signature setting (p > 1, q > 1) it is more natural to work with the Jordan normal form rather than just the eigenvalue structure.…”

Manifolds with commuting Jacobi operators

“…Taking the inner product with x, y, and w then yields, respectively b 2 = 0, c 2 = 0, and a 2 = 0, which completes the proof of Assertion (2).…”