|
Abstract:
|
n this dissertation, we start from the curvature tensor of the pseudo-
Riemannian manifold or the algebraic curvature tensor on a vector space with a
(possibly indefinite) scalar product. The duality, proportionality and orthogonal-
ity principles of Osserman tensors are studied as they are properties of curvature
tensors that are characteristic of Riemannian Osserman manifolds. The estab-
lished principles are generalized to the pseudo-Riemannian case and are observed
in two directions. On the one hand, we are interested whether these principles
follow from Osserman’s conditions, and on the other, to what extent Osserman’s
conditions are a consequence of established principles. Quasi-Clifford tensors are
introduced as a generalization of Clifford tensors, and then some sufficient condi-
tions are given under which the totally duality principle holds for quasi-Clifford
tensors, and an example of a pseudo-Riemannian Osserman tensor is presented
for which the duality principle does not hold. The theorem on the existence of
the algebraic curvature tensor for the given Jacobi operators is proved, which is
used to prove the results on the principle of proportionality. The principle of
orthogonality is devised as a new potential characterization of Riemannian Osser-
man tensors. Every Riemannian Jacobi-orthogonal tensor is an Osserman tensor,
while Clifford and two-root Riemannian Oserman tensors are Jacobi-orthogonal.
Generalizations of the orthogonality principle in the pseudo-Riemannian case are
presented, especially in the cases of small dimensions 3 and 4. |