Duality principle and special Osserman manifolds

Abstract

We investigate the connection between the duality principle and the Osserman condition in a pseudo-Riemannian setting. We prove that a connected pointwise two-leaves Osserman manifold of dimension n 5 is globally Osserman and investigate the relation between the special Osserman condition and the two-leaves Osserman one.

Quasi-special Osserman manifolds

“…The duality principle for Osserman curvature tensor works for every known example, however we failed to prove it in general. In our previous work [1,2,4] we gave the affirmative answer only for the conditions of small index (ν ≤ 1) or low dimension (n ≤ 4). In this text, we restrict our attention to small numbers of eigenvalues of the reduced Jacobi operator.…”

“…This is the reason why we devote our attention to the first nontrivial case, the diagonalizable Osserman curvature tensor whose reduced Jacobi operator has two distinct eigenvalues, and we call it a two-leaves Osserman (Definition 3). The duality principle holds for n ≤ 4 [1][2][3][4], but every connected pointwise two-leaves Osserman manifold is a globally Osserman for n > 4 [2,9]. This is why we put the problem into a pure algebraic concept with an algebraic Osserman curvature tensor instead of working with an Osserman manifold and associated tangent bundles.…”

“…This is why we have started investigating the duality principle for Osserman curvature tensor in a pseudo-Riemannian setting. In a pseudo-Riemannian setting, the implication (1) looks inaccurate, and therefore we corrected it in the following way [1,2,4].…”

The duality principle for Osserman algebraic curvature tensors

“…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].…”

On quasi-Clifford Osserman curvature tensors

“…Additionally, we have the following partial results. 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].…”