We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type (r, s) in a vector space of signature (p, q). We then use these examples to establish some results concerning higher order Osserman and higher order Jordan Osserman algebraic curvature tensors. §1 Introduction A 4 tensor R is said to be an algebraic curvature tensor if it satisfies the well known symmetries of the Riemannian curvature tensor, i.e.It is clear that the Riemann curvature tensor defines an algebraic curvature tensor at each point of the manifold. Conversely, every algebraic curvature tensor is geometrically realizable [10]. We remark that it is often convenient to study certain geometric problems in a purely algebraic setting.Let R be an algebraic curvature tensor on a vector space V of signature (p, q). The Jacobi operator J R is the self-adjoint linear map defined by:Here the natural domains of definition are the pseudo-spheres of unit timelike (−) and spacelike (+) vectors in V :2000 Mathematics Subject Classification. 53B20 Key words and phrases. Higher order Jacobi operator, Osserman algebraic curvature tensors, Jordan Osserman algebraic curvature tensors †Research partially supported by the NSF (USA) and the MPI (Leipzig) ‡Research partially supported by JSPS Post Doctoral Fellowship Program (Japan) and the MPI (Leipzig) Typeset by A M S-T E X Stanilov has extended J R to non-degenerate subspaces of arbitrary dimension. Let σ be a non-degenerate subspace of V . If {v i } is a basis for σ, then let h ij := (v i , v j ) describe the restriction of the metric on V to the subspace σ. Since σ is non-degenerate, the matrix (h ij ) is invertible and we let (h ij ) be the inverse. The higher order Jacobi operator is defined [16] by generalizing equation (1.1.a): J R (σ)y := ij h ij R(y, v i )v j ;2

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Let R be an algebraic curvature tensor on a vector space of signature (p, q) defining a spacelike Jordan Osserman Jacobi operator J R . We show that the eigenvalues of J R are real and that J R is diagonalizable if p < q.

We study the higher order Jacobi operator in pseudo-Riemannian geometry. We exhibit a family of manifolds so that this operator has constant Jordan normal form on the Grassmannian of subspaces of signature (r, s) for certain values of (r, s). These pseudo-Riemannian manifolds are new and nontrivial examples of higher order Osserman manifolds.

We construct a family of pseudo-Riemannian manifolds so that the skew-symmetric curvature operator, the Jacobi operator, and the Szabó operator have constant eigenvalues on their domains of definition. This provides new and non-trivial examples of Osserman, Szabó, and IP manifolds. We also study when the associated Jordan normal form of these operators is constant.