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Osserman pseudo-Riemannian manifolds of signature (2,2)

Part of: Local differential geometry Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Novica Blažić ,
Neda Bokan  and
Zoran Rakić
Novica Blažić
Affiliation:
University of Belgrade, Faculty of Mathematics, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
Neda Bokan
Affiliation:
University of Belgrade, Faculty of Mathematics, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
Zoran Rakić
Affiliation:
University of Belgrade, Faculty of Mathematics, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
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Abstract

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A pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.

MSC classification

Secondary: 53B30: Lorentz metrics, indefinite metrics 53C50: Lorentz manifolds, manifolds with indefinite metrics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 3 , December 2001 , pp. 367 - 396
Copyright
Copyright © Australian Mathematical Society 2001

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