The Jordan algebras of Riemann, Weyl and curvature compatible tensors

Jordan Szabó Algebraic Covariant Derivative Curvature Tensors

We show that if R is a Jordan Szabó algebraic covariant derivative curvature tensor on a vector space of signature (p, q), where q ≡ 1 mod 2 and p < q or q ≡ 2 mod 4 and p < q − 1, then R = 0. This algebraic result yields an elementary proof of the geometrical fact that any pointwise totally isotropic pseudo-Riemannian manifold with such a signature (p, q) is locally symmetric.

Algebraic Curvature Tensors for Indefinite Metrics Whose Skew-symmetric Curvature Operator Has Constant Jordan Normal Form

We classify the connected pseudo-Riemannian manifolds of signature (p, q) with q ≥ 5 so that at each point of M the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes and so that the skew-symmetric curvature operator is not nilpotent for at least one point of M .

Landsberg Curvature, S-curvature and Riemann Curvature

The Weyl Curvature Conjecture

In this paper we review Penrose’s Weyl curvature conjecture which states that the concept of gravitational entropy and the Weyl tensor is somehow linked, at least in a cosmological setting. We give a description of a certain entity constructed from the Weyl tensor, from the very early history of our universe until the present day. Inflation is an important mechanism in our early universe for ho...

The Weyl Algebras