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

Abstract

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.

Four-dimensional Osserman metrics of neutral signature

“…First note that, by Theorem 1.5, ᏹ has Type Ia. Results of [Blažić et al 2001] then show that ᏹ either has constant sectional curvature, is locally isometric to a complex space form, or is locally isometric to a paracomplex space form. Since the curvature tensor of a paracomplex space form of constant paraholomorphic sectional curvature κ satisfies R(x, y)z = 1 4 κ(R 0 (x, y)z − R J (x, y)z), this is ruled out by Theorem 1.5, thus proving Theorem 1.8.…”

“…There is another family of Osserman 4-manifolds with diagonalizable Jacobi operator, namely, the paracomplex space forms [Blažić et al 2001]. Although the geometry of complex and paracomplex space forms is very similar, the Jordan-Osserman condition distinguishes them.…”

“…It then follows from the discussion in [Blažić et al 2001; that there is an orthonormal basis Ꮾ such that the nonzero components of the curvature tensor are…”

“…Type III models. For M of this type, there exists by [Blažić et al 2001; an orthonormal basis {e 1 , e 2 , e 3 , e 4 } for V such that the nonvanishing components of A are It now follows that r (u) = 2 while r (v) ≤ 1 and hence M is not null Jordan Osserman. This completes the proof of Theorem 1.5.…”

“…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.…”