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

Abstract

The main result we give in this brief note relates, under suitable hypotheses, the ϕ-null Osserman, the null Osserman and the classical Osserman conditions to each other, via semi-Riemannian submersions as projection maps of principal torus bundles arising from a Lorentzian S-manifold.

An Osserman-type condition on g.f.f-manifolds with Lorentz metric

“…Since u = ξ 1 +x, x ∈ S ϕ (ξ 1 ), using (2), it is easy to see that the eigenvalues and the eigenvectors of the Jacobi operatorR u are connected with those of R x | x ⊥ ∩Im ϕ . Namely, one can prove that v ∈ x ⊥ ∩ Imϕ is an eigenvector of R x related to the eigenvalue λ if and only if it is a geometrically realized eigenvector ofR u related to the eigenvalue λ + 1 ( [5]). Now, let us fix p ∈ M and, following [16], identify S ϕ (ξ 1 ) ∼ = S 2n−1 .…”