We study when the Jacobi operator associated to the Weyl confor-mal 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… (More)
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.
We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian manifolds which are modeled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov-Petrova.
There is a 14-dimensional algebraic curvature tensor which is Jacobi–Tsankov (i.e. J (x)J (y) = J (y)J (x) for all x, y) but which is not 2-step Jacobi nilpotent (i.e. J (x)J (y) = 0 for some x, y); the minimal dimension where this is possible is 14. We determine the group of symmetries of this tensor and show that it is geometrically realizable by a wide… (More)