Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-04-30T12:00:25.563Z Has data issue: false hasContentIssue false
  • English
  • Français

Osserman pseudo-Riemannian manifolds of signature (2,2)

Part of: Global differential geometry Local differential geometry

Published online by Cambridge University Press:  09 April 2009

Novica Blažić ,
Neda Bokan  and
Zoran Rakić
Novica Blažić
Affiliation:
University of Belgrade, Faculty of Mathematics, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia e-mail: blazicn@matf.bg.ac.yu e-mail: neda@matf.bg.ac.yu e-mail: zrakic@matf.bg.ac.yu
Neda Bokan
Affiliation:
University of Belgrade, Faculty of Mathematics, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia e-mail: blazicn@matf.bg.ac.yu e-mail: neda@matf.bg.ac.yu e-mail: zrakic@matf.bg.ac.yu
Zoran Rakić
Affiliation:
University of Belgrade, Faculty of Mathematics, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia e-mail: blazicn@matf.bg.ac.yu e-mail: neda@matf.bg.ac.yu e-mail: zrakic@matf.bg.ac.yu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.

MSC classification

Secondary: 53B30: Lorentz metrics, indefinite metrics 53C50: Lorentz manifolds, manifolds with indefinite metrics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 3 , December 2001 , pp. 367 - 396
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Akivis, M. A. and Konnov, V. V., ‘Some local aspects of the theory of conformal structures’, Uspekhi Mat. Nauk 48 (1993), 3–40Google Scholar
(Russian), translation in Russian Math. Surveys 48 (1993), 1–35.Google Scholar
[2]Alekseevsky, D. M., Blažić, N., Bokan, N. and Rakić, Z., ‘Selfdual pointwise Osserman spaces’, Arch. Math. (Brno) 35 (1999), 193–201.Google Scholar
[3]Ambrose, W. and Singer, I. M., ‘A theorem on holonomy’, Trans. Arner Math. Soc. 75 (1953), 428–443.CrossRefGoogle Scholar
[4]Baros, M. and Romero, A., ‘Indefinite Kähler manifolds’, Math. Ann. 261 (1982), 55–62.CrossRefGoogle Scholar
[5]Blažié, N., Bokan, N. and Gilkey, P., ‘A note on Osserman Lorentzian manifolds’, Bull. London Math. Soc. 29 (1997), 227–230.CrossRefGoogle Scholar
[6]Blažié, N., Bokan, N. and Rakić, Z., ‘Foliation of a dynamically homogeneous neutral manifold’, J. Math. Phys. 39 (1998), 6118–6124.CrossRefGoogle Scholar
[7]Blažié, N., Bokan, N. and Rakić, Z., ‘A note on Osserman conjecture and isotropic covariant derivative of curvature’, Proc. Arner Math. Soc. 128 (2000), 245–253.CrossRefGoogle Scholar
[8]Boeckx, E., Kowalski, O. and Vanhecke, L., Riemannian manifolds of conullity two (World Scientific, Singapore, 1996).CrossRefGoogle Scholar
[9]Bonome, A., Castro, R., García-Rio, E., Hervella, L. and Vázquez-Lorenzo, R., ‘Non-symmetric Osserman indefinite Kähler manifolds’, Proc. Amer Math. Soc. 126 (1998), 2763–2769.CrossRefGoogle Scholar
[10]Chi, Q. S., ‘A curvature characterization of certain locally rank-one symmetric spaces’, J. Differential Geom. 28 (1988), 187–202.CrossRefGoogle Scholar
[11]Chi, Q. S., ‘Curvature characterization and classification of rank-one symmetric spaces’, Pacific J. Math. 150 (1991), 31–42.CrossRefGoogle Scholar
[12]Chi, Q. S., ‘Quaternionic Kähler manifolds and a curvature characterization of two-point homogeneous spaces’, Illinois J. Math. 35 (1991), 408–419.CrossRefGoogle Scholar
[13]Dajczer, M. and Nomizu, K., ‘On sectional curvature of indefinite metrics II’, Math. Ann. 247 (1980), 279–282.CrossRefGoogle Scholar
[14]Gadea, P. M. and Masqué, J. M., ‘Classification of non-flat para-Kählerian space forms’, Houston J. Math. 21 (1995), 89–94.Google Scholar
[15]Garciá-Rio, E., Kupeli, D. N. and V´zquez-Abal, M. E., ‘On a problem of Osserman in Lorentzian geometry’, Differential Geom. Appl. 7 (1997), 85–100.CrossRefGoogle Scholar
[16]García-Rio, E., V´zquez-Abal, M. E. and V´zquez-Lorenzo, R., ‘Nonsymmetric Osserman pseudoRiemannian manifolds’, Proc. Amer Math. Soc. 126 (1998), 2771–2778.CrossRefGoogle Scholar
[17]Gilkey, P. B., ‘Manifolds whose curvature operator has constant eigenvalues at the basepoint’, J. Geom. Anal. 4 (1994), 155–158.CrossRefGoogle Scholar
[18]Gilkey, P. B., Swann, A. and Vanhecke, L., ‘Isoparametric geodesic spheres and a conjecture of Osserman concerning the Jacobi operator’, Quart. J. Math. Oxford Ser. (2) 46 (1995), 299–320.CrossRefGoogle Scholar
[19]Kowalski, O., Tricerri, F. and Vanhecke, L., ‘Curvature homogeneous Riemannian manifolds’, J. Math. Pures Appl. 71 (1992), 471–501.Google Scholar
[20]Malcev, A. I., Foundations of linear algebra (Nauka, Moskva, 1970).Google Scholar
[21]O'Neill, B., Semi-Riemannian geometry with applications to relativity (Academic Press, New York, 1983).Google Scholar
[22]Osserman, R., ‘Curvature in the eighties’, Amer Math. Monthly 97 (1990), 731–756.CrossRefGoogle Scholar
[23]Rakić, Z., ‘Rank 2 symmetric Osserman spaces’, Bull. Austral. Math. Soc. 56 (1988), 517–521.CrossRefGoogle Scholar
[24]Rozenfeld, B. A., Non-Euclidean geometries (GITTL, Moskva, 1955).Google Scholar
[25]Ruse, H. S., Walker, A. G. and Willmore, T. J., Harmonic spaces (Edizioni Cremonese, Rome, 1961).Google Scholar
[26]Singer, I. M., ‘Infinitesimally homogeneous spaces’, Comm. Pure Appl. Math. 13 (1960), 685–697.CrossRefGoogle Scholar
[27]Vanhecke, L., ‘Scalar curvature invariants and local homogeneity’, Rend. Circ. Mat. Palermo 49 suppl. (1997), 275–287.Google Scholar
[28]Walker, A. G., ‘Canonical form for a Riemannian space with a parallel field of null planes’, Quart. J. Math. Oxford Ser. (2) 1 (1950), 69–79.CrossRefGoogle Scholar
[29]Wolf, J. A., Spaces of constant curvature (Univ. California, Berkeley, 1972).Google Scholar
[30]Wu, H., ‘Holonomy groups of indefinite metrics’, Pacific J. Math. 20 (1967), 351–392.CrossRefGoogle Scholar