Pseudo-Riemannian Jacobi–videv Manifolds

Abstract

Abstract. We exhibit several families of Jacobi-Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi-Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure.

Geometric Realizations of Para-Hermitian Curvature Models

“…Theorem 5.3). Furthermore, either the complex Jacobi operator or the complex curvature operator completely determine the components in W ⊥ 7 of a curvature tensor [8]; the algebraic condition determining W 7 also plays a role in the study of Jacobi-Ricci commuting curvature tensors [36].…”

Stanilov-Tsankov-Videv Theory

“…Similarly Jacobi-Videv and skew-Videv are equivalent conditions. This follows from the following result [20]: Theorem 1. Let M be a model and let T be a self-adjoint linear transformation of V .…”

“…The condition that M is pseudo-Einstein does not, however, imply that M is Jacobi-Videv in the higher signature setting as the following [20] shows:…”

“…Let {e i } be an orthonormal basis for V . Let {e We refer to [20] for the proof of the following result: Definition 9. Let M = (V, ·, · , R) be a model.…”

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

“…Let (M, g) be a pseudo-Riemannian manifold and denote by Ric and J the Ricci operator and the Jacobi operator of (M, g), respectivey. The manifold is said to be The class of Jacobi-Ricci commuting manifolds coincides with the one of curvature-Ricci commuting manifolds [14], which are also known in literature as Ricci semi-symmetric spaces. (We may refer to [17] for geometric interpretations and further references.)…”