We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also study similar questions related to the skew-symmetric curvature operator defined by the Weyl conformal curvature tensor.
We use curvature decompositions to construct generating sets for the space of algebraic curvature tensors and for the space of tensors with the same symmetries as those of a torsion free, Ricci symmetric connection; the latter naturally appear in relative hypersurface theory.
In 2005 Gilkey and Nikčević introduced complete -curvature homogeneous pseudo-Riemannian manifolds of neutral signature , which are -modeled on an indecomposable symmetric space, but which are not -curvature homogeneous. In this paper the authors continue their study of the same family of manifolds by examining their isometry groups and the isometry groups of their -models.
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