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

Blažić, Novica; Bokan, Neda; Rakić, Zoran

doi:10.1090/S0002-9939-99-05131-X

A note on the Osserman conjecture and isotropic covariant derivative of curvature
HTML articles powered by AMS MathViewer

by Novica Blažić, Neda Bokan and Zoran Rakić PDF

Proc. Amer. Math. Soc. 128 (2000), 245-253 Request permission

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.

References

Novica Blažić, Neda Bokan, and Peter Gilkey, A note on Osserman Lorentzian manifolds, Bull. London Math. Soc. 29 (1997), no. 2, 227–230. MR 1426003, DOI 10.1112/S0024609396002238
N. Blažić, N. Bokan, P. Gilkey and Z. Rakić, Pseudo-Riemannian Osserman manifolds, (preprint 1997).
N. Blažić, N. Bokan and Z. Rakić, Characterization of 4-dimensional Osserman pseudo-Riemannian manifolds, (preprints 1995, 1997).
—, Characterization of type II Osserman manifolds in terms of connection forms, (preprint 1997).
—, Recurrent spacelike (timelike) Osserman spaces, to appear, Bull. Acad. Serbe Sci. Arts Cl. Sci Math. Natur.
Novica Blažić, Neda Bokan, and Zoran Rakić, The first order PDE system for type III Osserman manifolds, Publ. Inst. Math. (Beograd) (N.S.) 62(76) (1997), 113–119. MR 1488635
A. Bonome, R. Castro, E. García-Río, L. Hervella, and R. Vázquez-Lorenzo, Nonsymmetric Osserman indefinite Kähler manifolds, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2763–2769. MR 1476121, DOI 10.1090/S0002-9939-98-04659-0
Peter Bueken and Lieven Vanhecke, Examples of curvature homogeneous Lorentz metrics, Classical Quantum Gravity 14 (1997), no. 5, L93–L96. MR 1448278, DOI 10.1088/0264-9381/14/5/004
Quo-Shin Chi, A curvature characterization of certain locally rank-one symmetric spaces, J. Differential Geom. 28 (1988), no. 2, 187–202. MR 961513
E. García-Río, D. N. Kupeli, and M. E. Vázquez-Abal, On a problem of Osserman in Lorentzian geometry, Differential Geom. Appl. 7 (1997), no. 1, 85–100. MR 1441921, DOI 10.1016/S0926-2245(96)00037-X
E. García-Río, M. E. Vázquez-Abal, and R. Vázquez-Lorenzo, Nonsymmetric Osserman pseudo-Riemannian manifolds, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2771–2778. MR 1476128, DOI 10.1090/S0002-9939-98-04666-8
E. García-Rio, D. N. Kupeli, M. E. Vázquez-Abal, R. Vázquez-Lorenzo, Osserman affine connections and their Riemann extensions, preprint, 1997.
Peter B. Gilkey, Manifolds whose curvature operator has constant eigenvalues at the basepoint, J. Geom. Anal. 4 (1994), no. 2, 155–158. MR 1277503, DOI 10.1007/BF02921544
Peter Gilkey, Andrew Swann, and Lieven Vanhecke, Isoparametric geodesic spheres and a conjecture of Osserman concerning the Jacobi operator, Quart. J. Math. Oxford Ser. (2) 46 (1995), no. 183, 299–320. MR 1348819, DOI 10.1093/qmath/46.3.299
Andreas Koutras and Colin McIntosh, A metric with no symmetries or invariants, Classical Quantum Gravity 13 (1996), no. 5, L47–L49. MR 1390083, DOI 10.1088/0264-9381/13/5/002
Robert Osserman, Curvature in the eighties, Amer. Math. Monthly 97 (1990), no. 8, 731–756. MR 1072814, DOI 10.2307/2324577
R. Osserman and P. Sarnak, A new curvature invariant and entropy of geodesic flows, Invent. Math. 77 (1984), no. 3, 455–462. MR 759262, DOI 10.1007/BF01388833
Friedbert Prüfer, Franco Tricerri, and Lieven Vanhecke, Curvature invariants, differential operators and local homogeneity, Trans. Amer. Math. Soc. 348 (1996), no. 11, 4643–4652. MR 1363946, DOI 10.1090/S0002-9947-96-01686-8
H. S. Ruse, A. G. Walker, and T. J. Willmore, Harmonic spaces, Consiglio Nazionale delle Ricerche Monografie Matematiche, vol. 8, Edizioni Cremonese, Rome, 1961. MR 0142062
I. M. Singer, Infinitesimally homogeneous spaces, Comm. Pure Appl. Math. 13 (1960), 685–697. MR 131248, DOI 10.1002/cpa.3160130408
Shlomo Sternberg, Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0193578
L. Vanhecke, Scalar curvature invariants and local homogeneity, Rend. Circ. Mat. Palermo 49 suppl. (1997), 275–287.