Lorentzian three-manifolds with special curvature operators

Abstract

Three-dimensional Lorentzian manifolds whose skew-symmetric curvature operators have constant eigenvalues are studied. A complete algebraic description is given, which allows a complete characterization at the differentiable level of manifolds which additionally are assumed to be locally symmetric or homogeneous.

Three-dimensional Ivanov–Petrova manifolds

“…We now prove that Lorentzian metrics described in Theorem 3.3 do not admit a parallel degenerate line field and, so, are not included in the examples given in Ref. 10. In fact, suppose that there exists a parallel degenerate line field with respect to a Lorentzian g described by ͑3.1͒ and ͑3.15͒.…”

“…10. If such an algebraic curvature tensor has no constant sectional curvature, then either its Ricci operator Q is diagonalizable and has rank 1 or Q is two-step nilpotent.…”

“…15 An IP algebraic curvature tensor in a three-dimensional Lorentzian vector space, if not of constant sectional curvature, has a Ricci operator Q either diagonalizable and of rank 1 or two-step nilpotent. 10 In order to find examples, we consider on an open subset of R 3 ͓w , x , y͔ a Lorentzian metric,…”

“…We briefly recall that a parallel degenerate line field is a one-dimensional distribution D over a Lorentzian manifold ͑M , g͒, such that ٌD ʚ D. 10 It is locally spanned by a nonvanishing null vector field u, satisfying ٌu = u, where is a ͑local͒ 1-form over M. In particular, if = 0, then u is a parallel null vector field.…”

Algebraic Properties of Curvature Operators in Lorentzian Manifolds with Large Isometry Groups

“…Recently, IP manifolds have been extended and largely investigated in pseudo-Riemannian settings (see [9,12,16] and references therein).…”