Four-dimensional Osserman metrics of neutral signature

Eduardo García-Río, Peter B. Gilkey, M. Elena Vázquez-Abal and Ramón Vázquez-Lorenzo

Vol. 244 (2010), No. 1, 21–36

DOI: 10.2140/pjm.2010.244.21

Abstract

In the algebraic context, we show null Osserman, spacelike Osserman, and timelike Osserman are equivalent conditions for a model of signature (2,2). We also classify the null Jordan Osserman models of signature (2,2). In the geometric context, we show that a pseudo-Riemannian manifold with this signature is null Jordan Osserman if and only if either it has constant sectional curvature or it is locally a complex space form.

Keywords

spacelike, timelike and null Jacobi operator, Osserman and Jordan Osserman metric, neutral signature (2,2)

Mathematical Subject Classification 2000

Primary: 53C20, 53C50

Secondary: 53B30

Milestones

Received: 3 April 2008

Revised: 15 September 2009

Accepted: 24 September 2009

Published: 17 November 2009