Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
higher order Jacobi operator; Osserman algebraic curvature tensors; Jordan Osserman algebraic curvature tensors
higher order Jacobi operator; Osserman algebraic curvature tensors; Jordan Osserman algebraic curvature tensors
Summary:
We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type $(r,s)$ in a vector space of signature $(p,q)$. We then use these examples to establish some results concerning higher order Osserman and higher order Jordan Osserman algebraic curvature tensors.
References:
[1] Blažić N., Bokan N., Gilkey P.: A note on Osserman Lorentzian manifolds. Bull. London Math. Soc. 29 (1997), 227-230. MR 1426003
[2] Blažić N., Bokan N., Gilkey P., Rakić Z.: Pseudo-Riemannian Osserman manifolds. J. Balkan Society of Geometers l2 (1997), 1-12. MR 1662081
[3] Bonome A., Castro R., García-Río E., Hervella L., Vázquez-Lorenzo R.: Nonsymmetric Osserman indefinite Kähler manifolds. Proc. Amer. Math. Soc. 126 (1998), 2763-2769. MR 1476121
[4] Chi Q.-S.: A curvature characterization of certain locally rank-one symmetric spaces. J. Differential Geom. 28 (1988), 187-202. MR 0961513 | Zbl 0654.53053
[5] Dotti I., Druetta M.: Negatively curved homogeneous Osserman spaces. Differential Geom. Appl. 11 (1999), 163-178. MR 1712119 | Zbl 0970.53031
[6] García-Rió E., Kupeli D., Vázquez-Abal M.E.: On a problem of Osserman in Lorentzian geometry. Differential Geom. Appl. 7 (1997), 85-100. MR 1441921
[7] García-Rió E., Vázquez-Abal M.E., Vázquez-Lorenzo R.: Nonsymmetric Osserman pseudo-Riemannian manifolds. Proc. Amer. Math. Soc. 126 (1998),2771-2778. MR 1476128
[8] Gilkey P.: Manifolds whose curvature operator has constant eigenvalues at the basepoint. J. Geom. Anal. 4 (1994), 155-158. MR 1277503 | Zbl 0797.53010
[9] Gilkey P.: Algebraic curvature tensors which are $p$ Osserman. to appear in Differential Geom. Appl. MR 1836275 | Zbl 1031.53034
[10] Gilkey P.: Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor. World Scientific, 2002. MR 1877530 | Zbl 1007.53001
[11] Gilkey P., Ivanova R.: The Jordan normal form of Osserman algebraic curvature tensors. Results Math. 40 (2001), 192-204. MR 1860368 | Zbl 0999.53014
[12] Gilkey P., Stavrov I.: Curvature tensors whose Jacobi or Szabó operator is nilpotent on null vectors. Bull. London Math. Soc., to appear. MR 1924351 | Zbl 1043.53018
[13] Gilkey P., Stanilov G., Videv V.: Pseudo-Riemannian manifolds whose generalized Jacobi operator has constant characteristic polynomial. J. Geom. 62 (1998), 144-153. MR 1631494 | Zbl 0906.53046
[14] Gilkey P., Swann A., Vanhecke L.: Isoparametric geodesic spheres and a conjecture of Osserman regarding the Jacobi operator. Quart. J. Math. Oxford Ser. 46 (1995), 299-320. MR 1348819
[15] Osserman R.: Curvature in the eighties. Amer. Math. Monthly 97 (1990), 731-756. MR 1072814 | Zbl 0722.53001
[16] Stanilov G.: Curvature operators based on the skew-symmetric curvature operator and their place in Differential Geometry. preprint, 2000.
[17] Stanilov G., Videv V.: On Osserman conjecture by characteristical coefficients. Algebras Groups Geom. 12 (1995), 157-163. MR 1325979 | Zbl 0827.53042
[18] Stanilov G., Videv V.: Four-dimensional pointwise Osserman manifolds. Abh. Math. Sem. Univ. Hamburg 68 (1998), 1-6. MR 1658408 | Zbl 0980.53058
Search
Partner of
