Curvature Tensors Whose Jacobi or Szabó Operator Is Nilpotent on Null Vectors

Abstract

We show that any k Osserman Lorentzian algebraic curvature tensor has constant sectional curvature and give an elementary proof that any local 2 point homogeneous Lorentzian manifold has constant sectional curvature. We also show that a Szabó Lorentzian covariant derivative algebraic curvature tensor vanishes.

The Structure of Algebraic Covariant Derivative Curvature Tensors

“…∇R = 0. This result was subsequently extended to the Lorentzian case [16]. In the higher signature setting, again the situation is unclear.…”

Nilpotent Szabó, Osserman and Ivanov-Petrova pseudo-Riemannian manifolds

“…If (M, g) is Szabó nilpotent of order 1, then S(x) = 0 for all x ∈ T M . This implies [14] that ∇R = 0 so (M, g) is a local symmetric space; this is to be regarded, therefore, as a trivial case. Gilkey, Ivanova, and Zhang [12] have constructed pseudo-Riemannian manifolds of any signature (p, q) with p ≥ 2 and q ≥ 2 which are Szabó nilpotent of order 2; these were the only previously known examples of Szabó manifolds which were not local symmetric spaces.…”

On twisted Riemannian extensions associated with Szab o metrics

“…He used this observation to prove that any two point homogeneous space is either at or is a rank one symmetric space. Subsequently Gilkey and Stravrov [9] extended this result to show that any Szabó Lorentzian manifold has constant sectional curvature. However, for metrics of higher signature the situation is dierent.…”