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A note on the Osserman conjecture and isotropic covariant derivative of curvature


Authors: Novica Blazic, Neda Bokan and Zoran Rakic
Journal: Proc. Amer. Math. Soc. 128 (2000), 245-253
MSC (1991): Primary 53B30, 53C50
DOI: https://doi.org/10.1090/S0002-9939-99-05131-X
Published electronically: May 11, 1999
MathSciNet review: 1641649
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Abstract: Let $M$ be a Riemannian manifold with the Jacobi operator, which has constant eigenvalues, independent on the unit vector $X\in T_{p}M$ and the point $p\in M$. Osserman conjectured that these manifolds are flat or rank-one locally symmetric spaces ($\nabla R =0$). It is known that for a general pseudo-Riemannian manifold, the Osserman-type conjecture is not true and 4-dimensional Kleinian Jordan-Osserman manifolds are curvature homogeneous. We show that the length of the first covariant derivative of the curvature tensor is isotropic, i.e. $\Vert \nabla R\Vert =0$. For known examples of 4-dimensional Osserman manifolds of signature $(--++)$ we check also that $\Vert \nabla R\Vert =0$. By the presentation of a class of examples we show that curvature homogeneity and $\Vert \nabla R\Vert =0$ do not imply local homogeneity; in contrast to the situation in the Riemannian geometry, where it is unknown if the Osserman condition implies local homogeneity.


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  • [1] N. Bla\v{z}i\'{c}, N. Bokan and P. Gilkey, A Note on Osserman Lorentzian manifolds, Bull. London Math. Soc. 29 (1997), 227-230. MR 97m:53111
  • [2] N. Bla\v{z}i\'{c}, N. Bokan, P. Gilkey and Z. Raki\'{c}, Pseudo-Riemannian Osserman manifolds, (preprint 1997).
  • [3] N. Bla\v{z}i\'{c}, N. Bokan and Z. Raki\'{c}, Characterization of 4-dimensional Osserman pseudo-Riemannian manifolds, (preprints 1995, 1997).
  • [4] -, Characterization of type II Osserman manifolds in terms of connection forms, (preprint 1997).
  • [5] -, Recurrent spacelike (timelike) Osserman spaces, to appear, Bull. Acad. Serbe Sci. Arts Cl. Sci Math. Natur.
  • [6] -, The first order PDE system for type III Osserman manifolds, Publ. de l'Inst. Math. 62(76) (1997), 113-119. MR 98m:53029
  • [7] A. Bonome, R. Castro, E. Garcia-Rio, L. M. Hervella, R. Vázquez-Lorenzo, Nonsymmetric Osserman indefinite Kähler manifolds, Proc. Amer. Math. Soc. 126 (1998), 2763-2769. MR 98m:53088
  • [8] P. Bueken, L. Vanhecke, Examples of curvature homogeneous Lorentz metrics, Class. Quantum Grav. 14 (1997), L93-L96. MR 98c:53073
  • [9] Q. S. Chi, A curvature characterization of certain locally rank-one symmetric spaces, J. Diff. Geom. 28 (1988), 187-202. MR 90a:53060
  • [10] E. García-Rio, D. N. Kupeli and M. E. Vázquez-Abal, On a problem of Osserman in Lorentzian geometry, Differential Geometry and its Applications 7 (1997), 85-100. MR 98a:53099
  • [11] E. García-Rio, M. E. Vázquez-Abal and R. Vázquez-Lorenzo, Nonsymmetric Osserman pseudo Riemannian manifolds, Proc. Amer. Math. Soc. 126 (1998), 2771-2778. MR 98m:53089
  • [12] E. García-Rio, D. N. Kupeli, M. E. Vázquez-Abal, R. Vázquez-Lorenzo, Osserman affine connections and their Riemann extensions, preprint, 1997.
  • [13] P. Gilkey, Manifolds whose curvature operator has constant eigenvalues at the basepoint, J. Geom. Anal. 4 (1994), 155-158. MR 95f:53084
  • [14] P. Gilkey, A. Swann and L. Vanhecke, Isoparametric geodesic spheres and a conjecture of Osserman concerning the Jacobi operator, Quart. J. Math. Oxford. 46 (1995), 299-320. MR 96h:53051
  • [15] A. Koutras, C. Mcintosh, A metric with no invariants, Class. Quantum Grav. 13 (1996), L47-9. MR 97g:83027
  • [16] R. Osserman, Curvature in the eighties, Amer. Math. Monthly 97 (1990), 731-756. MR 91i:53001
  • [17] R. Osserman and P. Sarnak, A new curvature invariant and entropy of geodesic flows, Invent. Math. 77 (1984), 455-462. MR 86a:58054
  • [18] F. Prüfer, F. Tricerri, L. Vanhecke, Curvature invariants, differential operators and local homogeneity, Trans. Am. Math. Soc. 348 (1996), 4643-4652. MR 97a:53074
  • [19] H. S. Ruse, A. G. Walker, T. J. Willmore, Harmonic spaces, Cremonese, Rome, 1961. MR 25:5456
  • [20] I. M. Singer, Infinitesimally homogeneous spaces, Commun. Pure Appl. Math. 13 (1960), 685-697. MR 24:A1100
  • [21] S. Sternberg, Lectures on Differential Geometry,, Prentice-Hall, Englewood Cliffs, NJ, 1964. MR 33:1797
  • [22] L. Vanhecke, Scalar curvature invariants and local homogeneity, Rend. Circ. Mat. Palermo 49 suppl. (1997), 275-287. CMP 98:09

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Additional Information

Novica Blazic
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia
Email: blazicn@matf.bg.ac.yu

Neda Bokan
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia
Email: neda@matf.bg.ac.yu,

Zoran Rakic
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia
Email: zrakic@matf.bg.ac.yu

DOI: https://doi.org/10.1090/S0002-9939-99-05131-X
Keywords: Pseudo-Riemannian manifold, curvature tensor, Jacobi operator, Kleinian Osserman spacelike (timelike) manifold, Osserman conjecture, isotropicity
Received by editor(s): November 6, 1997
Received by editor(s) in revised form: March 3, 1998
Published electronically: May 11, 1999
Additional Notes: Research partially supported by SFS, Project #04M03.
Communicated by: Christopher Croke
Article copyright: © Copyright 1999 American Mathematical Society

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