Curvature properties of φ-null Osserman Lorentzian S-manifolds

Abstract

We expound some results about the relationships between the Jacobi operators with respect to null vectors on a Lorentzian S-manifold and the Jacobi operators with respect to particular spacelike unit vectors. We study the number of the eigenvalues of such operators on Lorentzian S-manifolds satisfying the -null Osserman condition, under suitable assumptions on the dimension of the manifold. Then, we provide in full generality a new curvature characterization for Lorentzian S-manifolds and we use it to obtain an algebraic decomposition for the Riemannian curvature tensor of -null Osserman Lorentzian S-manifolds. MSC:53C50, 53C15, 53B30, 53C25

Principal torus bundles of Lorentzian ${\mathcal S}$-manifolds and the φ-null Osserman condition

“…If z ∈ T p M is a unit timelike vector, the null congruence set of z is defined to be the set Following [7] and [8], we recall the basic facts related with the definition of the ϕ-null Osserman condition.…”

“…Since we can write u = (ξ 1 ) p + x, with x ∈ S ϕ ((ξ 1 ) p ), there is a natural one-to-one correspondence between the two kinds of Jacobi operator R x : x ⊥ → x ⊥ andR u :ū ⊥ →ū ⊥ . In [8] it is provided the relationship between these two operators with respect to the ϕ-null Osserman condition, which we summarize in the following proposition. The above result enables us to write the definition of the ϕ-null Osserman condition in terms of operator R x , x ∈ S ϕ ((ξ 1 ) p ), instead ofR u , u ∈ N ϕ ((ξ 1 ) p ).…”

“…This was motivated by the following considerations: although any Lorentzian Sasaki manifold (M, ϕ, ξ, η, g) with constant ϕ-sectional curvature is globally null Osserman with respect to the timelike vector field ξ, there is no similar result when we consider Lorentzian S-manifolds, which generalize Lorentzian Sasaki ones, and moreover, as we proved in [8], no Lorentzian S-manifold can be neither null Osserman, nor Osserman. Further basic properties of such manifolds are studied in [7] and developed in [8]. We refer the reader to both works for more details about the ϕ-null Osserman condition, whilst the general reference for the whole Osserman framework is [20].…”

“…[7,8]). A Lorentzian g.f.f -manifold (M, ϕ, ξ α , η α , g) is said to be ϕ-null Osserman with respect to (ξ 1 ) p , p ∈ M , if the eigenvalues ofR u and their multiplicities are independent of u ∈ N ϕ ((ξ 1 ) p ).…”

“…[8]). Let (M, ϕ, ξ α , η α , g) be a Lorentzian S-manifold, dim(M ) = 2n + s, s For any p ∈ M , M is ϕ-null Osserman with respect to (ξ 1 ) p if and only if the eigenvalues of R x with their multiplicities are independent of x ∈ S ϕ ((ξ 1 ) p ).…”