On some aspects of duality principle

Abstract

This paper is devoted to the study of the relation between Osserman algebraic curvature tensors and algebraic curvature tensors which satisfy the duality principle. We give a short overview of the duality principle in Osserman manifolds and extend this notion to null vectors. Here, it is proved that a Lorentzian totally Jacobi-dual curvature tensor is a real space form. Also, we find out that a Clifford curvature tensor is Jacobi-dual. We provide a few examples of Osserman manifolds which are totally Jacobidual and an example of an Osserman manifold which is not totally Jacobi-dual.

The duality principle for Osserman algebraic curvature tensors

“…Recently, for algebraic curvature tensors in the Riemannian signature, it was shown that the duality principle is in fact equivalent to the Osserman property (in [5] for n ≤ 4, and in [14] for all n). In [4] this equivalence was extended to the Lorentzian signature, and it was also shown that the duality principle holds for algebraic curvature tensors R with the Clifford structure (all such R are Osserman). Further results on the duality principle can be found in [1,2].…”

“…The natural environment for using the duality principle for null vectors is Osserman manifolds of the neutral signature (p, p), by [8,Theorem 1.2]. In the signature (2, 2) there exists a manifold which is Osserman, but which does not satisfy the duality principle for null vectors, see [4,Section 6].…”

“…The reason for excluding null vectors is explained in [3,Remark 4] and [4,Section 6]. We have the following theorem.…”

On quasi-Clifford Osserman curvature tensors

“…Any four-dimensional Osserman R is Jacobi-dual, see Andrejić [1]. Any Lorentzian totally Jacobidual R has constant sectional curvature, see Andrejić and Rakić [4]. Any fourdimensional Jacobi-dual R such that J X is diagonalizable for some nonnull X is Osserman, see Andrejić [2].…”

“…In the Riemannian setting (g is positive definite), one of important features of an Osserman algebraic curvature tensor is the duality principle, given by Rakić [12]. Generalizations to a pseudo-Riemannian setting (see Andrejić and Rakić [3,4]) are possible by the following implication, Y is an eigenvector of J X =⇒ X is an eigenvector of J Y .…”