We show that any k Osserman Lorentzian algebraic curvature tensor has constant sectional curvature and give an elementary proof that any local 2 point homogeneous Lorentzian manifold has constant sectional curvature. We also show that a Szabó Lorentzian covariant derivative algebraic curvature tensor vanishes.

We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octonionic projective plane; we discuss these manifolds in the later sections of the paper.2000 Mathematics Subject Classification. 53C30 (Primary) 53C35 (Secondary). Key words and phrases. octonionic projective plane, para-octonionic projective plane, curvature tensor.

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.Since A λ = Aλ, E λ = Eλ. Both A and A λ preserve each generalized eigenspace E λ . The operator A is said to be Jordan simple if A λ = 0 on E λ for all λ. We say λ is an eigenvalue of A if dim{E λ } > 0; the spectrum Spec {A} ⊂ C is the set of complex eigenvalues of A.Lemma 1.1. Let A be a self-adjoint linear map of a vector space of signature (p, q).(1) We may decompose V = ⊕ Im (λ)≥0 E λ .(2) We have E λ ⊥ E µ if λ = µ and λ =μ.(3) The spaces E λ inherit non-degenerate metrics of signature (p λ , q λ ). (4) If λ / ∈ R, then p λ = q λ .

We show the existence of a modified Cliff(1, 1)-structure compatible with an Osserman 0-model of signature (2, 2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2, 2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman manifolds. DedicationThis paper is one of several projects that were begun by Novica Blažić but not completed owing to his untimely death in 2005. The work has been finished to preserve his mathematical legacy and is dedicated to his memory.gives rise to an Osserman 0-model on V .Proof. In the computation which follows we will use π x to denote the linear mapThe Jacobi operator corresponding to an algebraic curvature tensor of the form R Φ (see (1.a)) takes the formThe matrix representations of the operators J i (x) := J RΦ i (x) = 3π Φix , J ij (x) := J R (Φ i −Φ j ) (x) = 3π (Φix−Φj x)

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