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Pseudo Riemannian manifolds whose generalized Jacobi operator has constant characteristic polynomial. (English) Zbl 0906.53046
Let \(g\) be a nondegenerate inner product of signature \((p,q)\) on \(\mathbb{R}^m\). Let \(Gr_{r,s} (g)\) be the Grassmannian of \((r+s)\)-planes \(\pi\) such that \(g\) is nondegenerate of signature \((r,s)\) on \(\pi\). If \(R\) is an algebraic curvature tensor on \(\mathbb{R}^m\), define a generalized Jacobi operator on \(Gr_{r,s} (g)\) as follows. Let \(\{Y_i\}\) be a basis for \(\pi\in Gr_{r,s} (g)\) and \((g^{ij})\) the inverse of \((g_{ij}) =g(Y_i,Y_j)\). Then, the generalized Jacobi operator \(J_R(\pi)\) is defined by \[ J_R(\pi): Y\mapsto \sum_{1\leq i, j\leq r+s} g^{ij} R(Y,Y_i) Y_j. \] The curvature tensor \(R\) is said to be Osserman of type \((r,s)\) if the characteristic polynomial of \(J_R\) is constant on \(Gr_{r,s} (g)\).
The main purpose of this paper is to investigate the relation among the different kinds of Osserman conditions. In doing this, the authors show that an Osserman curvature tensor of type \((r,s)\) is 2-stein provided that \(2r+2 s\neq m\). Moreover, some more precise results are obtained for specified curvature tensors as well as for Riemannian and Lorentzian metrics.

MSC:
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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