Abstract
A complete description of Osserman four-manifolds whose Jacobi operators have a nonzero double root of the minimal polynomial is given.
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Alekseevsky, D., Blažić, N., Bokan, N., and Rakić, Z. Self-duality and pointwise Osserman manifolds,Arch. Math. (Brno) 35, 193–201, (1999).
Blažić, N., Bokan, N., and Gilkey, P. A note on Osserman Lorentzian manifolds,Bull. London Math. Soc. 29, 227–230, (1997).
Blažić, N., Bokan, N., and Rakić, Z. Osserman pseudo-Riemannian manifolds of signature (2,2),J. Aust. Math. Soc. 71, 367–395, (2001).
Blažić, N. and Gilkey, P. Curvature structure of self-dual 4-manifolds, to appear.
Bonome, A., Castro, R., García-Río, E., Hervella, L., and Vázquez-Lorenzo, R. Pseudo-Riemannian manifolds with simple Jacobi operators,J. Math. Soc. Japan 54, 847–875, (2002).
Bryant, R.L. Bochner-Kähler metrics,J. Amer. Math. Soc. 14, 623–715, (2001).
Chaichi, M., García-Río, E., and Matsushita, Y. Curvature properties of four-dimensional Walker metrics,Class. Quantum Grav. 22, 559–577, (2005).
Chi, Q.S. A curvature characterization of certain locally rank-one symmetric spaces,J. Diff. Geom. 28, 187–202, (1988).
Cruceanu, V., Fortuny, P., and Gadea, P.M. A survey on paracomplex geometry,Rocky Mount. J. Math. 26, 83–115. (1996).
Díaz-Ramos, J. C., García-Río, E., and Vázquez-Lorenzo, R. New examples of Osserman metrics with nondiagonalizable Jacobi operators,Differential Geom. Appl., to appear.
García-Río, E., Kupeli, D.N., and Vázquez-Abal, M.E. On a problem of Osserman in Lorentzian geometry,Differential Geom. Appl. 7, 85–100, (1997).
García-Río, E., Kupeli, D. N., and Vázquez-Lorenzo, R. Osserman manifolds in semi-Riemannian geometry,Lect. Notes Math. 1777, Springer-Verlag, Berlin, Heidelberg, New York, (2002).
García-Río, E., Vázquez-Abal, M.E., and Vázquez-Lorenzo, R. Nonsymmetric Osserman pseudo-Riemannian manifolds,Proc. Amer. Math. Soc. 126, 2771–2778, (1998).
García-Río, E., Vázquez-Abal, M.E., and Rakić, Z. Four-dimensional indefinite Kähler Osserman manifolds,J. Math. Phys. 46, 073505, (2005).
García-Río, E. and Vázquez-Lorenzo, R. Four-dimensional Osserman symmetric spaces,Geom. Dedicata 88, 147–151, (2001).
Gilkey, P.Geometric Properties of Natural Operators Defined by the Riemannian Curvature Tensor, World Scientific Publishing Co., Inc., River Edge, NJ, (2001).
Gilkey, P. and Ivanova, R. Spacelike Jordan-Osserman algebraic curvature tensors in the higher signature setting,Differential Geom. Valencia, (2001), 179–186, World Science Publishing, River Edge, NJ, (2002).
Gilkey, P. and Ivanova, R. The Jordan normal form of Osserman algebraic curvature tensors,Results Math. 40, 192–204, (2001).
Gilkey, P., Swann, A., and Vanhecke, L. Isoparametric geodesic spheres and a conjecture of Osserman concerning the Jacobi operator.Quart. J. Math. Oxford Ser. 46(2), 299–320, (1995).
Hitchin, N. Hypersymplectic quotients.Acta Acad. Sci. Tauriensis 124, supl., 169–180. (1990).
Magid, M.A. Shape operators of Einstein hypersurfaces in indefinite space forms,Proc. Amer. Math. Soc. 84, 237–242, (1982).
Nikolayevsky, Y. Osserman manifolds of dimension 8,Manuscripta Math. 115, 31–53, (2004).
Nikolayevsky, Y. Osserman conjecture in dimension ∈ 8, 16,Math. Ann. 331, 505–522, (2005).
Walker, A.G. Canonical form for a Riemannian space with a parallel field of null planes.Quart. J. Math. Oxford 1(2), 69–79, (1950).
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Díaz-Ramos, J.C., García-Río, E. & Vázquez-Lorenzo, R. Four-dimensional Osserman metrics with nondiagonalizable Jacobi operators. J Geom Anal 16, 39–52 (2006). https://doi.org/10.1007/BF02930986
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DOI: https://doi.org/10.1007/BF02930986