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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
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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Additional Information

Novica Blazic
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia

Neda Bokan
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia

Zoran Rakic
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia

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