zbMATH — the first resource for mathematics

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.

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C55 Global differential geometry of Hermitian and Kählerian manifolds
Full Text: DOI
[1] M. Baros andA. Romero,Indefinite K?hler manifolds. Math. Ann.261 (1982), 55-62. · Zbl 0487.53021
[2] N. Bla?i?, N. Bokan andP. Gilkey,A Note on Osserman Lorentzian manifolds, Bull. London Math. Soc.29 (1997), 227-230.
[3] N.Bla?i?, N.Bokan and Z.Raki?,Characterization of 4-dimensional Osserman pseudo-Riemannian manifolds, preprints (1995, 1997).
[4] A.Bonome, R.Castro, E.Garcia-rio, L.Hervella and R.V?squez-lorenzo,On Osserman semi-Riemannian manifolds, preprint (1997).
[5] A.Bonome, E.Garcia-Rio, L.Hervella and R.V?squez-Lorenzo,Nonsymmetric Osserman indefinite Kahler manifolds, preprint (1997).
[6] Q. S. Chi,A curvature characterization of certain locally rank one symmetric spaces, J. Diff. Geo.28 (1988), 187-202. · Zbl 0654.53053
[7] I.Dotti and M.Druetta,Negatively curved homogeneous Osserman spaces, preprint. · Zbl 0970.53031
[8] P. M. Gadea andJ. M. Masqu?,Classification of non-flat para-K?hlerian space forms, Houston J. Math.21 (1995), 89-94. · Zbl 0853.53027
[9] E. Garcia-Rio, D. N. Kupeli andM. E. V?zquez-Abal,On a problem of Osserman in Lorentzian geometry, Differential Geometry and its Applications7 (1997), 85-100. · Zbl 0880.53017
[10] E.Garcia-Rio, M. E.V?zquez-Abal and R.V?zquez-Lorenzo,Nonsymmetric Osserman pseudo Riemannian manifolds, preprint (1996).
[11] P. Gilkey,Manifolds whose curvature operator has constant eigenvalues at the base-point, J. Geo. Anal.4 (1994), 155-158. · Zbl 0797.53010
[12] P.Gilkey,Generalized Osserman Manifolds, Abh. Math. Sem. Univ. Hamburg, to appear. · Zbl 0980.53059
[13] P. Gilkey, A. Swann andL. Van Hecke,Isoparametric geodesic spheres and a conjecture of Osserman regarding the Jacobi Operator, Quart J Math46 (1995), 299-320. · Zbl 0848.53023
[14] R. Osserman,Curvature in the eighties, Amer. Math. Monthly97 (1990), 731-756. · Zbl 0722.53001
[15] I.Petrova and G.Stanilov,A generalized Jacobi operator on the 4dimensional Riemannian geometry, Annuare de l’Universite de Sofia ?St. Kliment Ohridski?85 (1991).
[16] G.Stanilov and V.Videv,Four dimensional pointwise Osserman manifolds, Abh. Math. Sem. Univ. Hamburg, to appear. · Zbl 0980.53058
[17] J. A. Wolf,Spaces of constant curvature, Univ. California (Berkeley, 1972). · Zbl 0234.33012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.