Geodesic spheres and two-point homogeneous spaces
In the Osserman conjecture and in the isoparametric conjecture it is stated that two-point homogeneous spaces may be characterized via the constancy of the eigenvalues of the Jacobi operator or the shape operator of geodesic spheres, respectively. These conjectures remain open, but in this paper we give complete positive results for similar statements about other symmetric endomorphism fields on small geodesic spheres. In addition, we derive more characteristic properties for this class of spaces by using other properties of small geodesic spheres. In particular, we study Riemannian manifolds with (curvature) homogeneous geodesic spheres.
KeywordsRiemannian Manifold Shape Operator Jacobi Operator Einstein Space Geodesic Sphere
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- [GSV] P. B. Gilkey, A. Swann and L. Vanhecke,Isoparametric geodesic spheres and a conjecture of Osserman concerning the Jacobi operator, The Quarterly Journal of Mathematics. Oxford, to appear.Google Scholar
- [Sz3] Z. I. Szabó,Spectral geometry for operator families on Riemannian manifolds, Proceedings of Symposia in Pure Mathematics54 (1993), 615–665.Google Scholar
- [TV2] F. Tricerri and L. Vanhecke,Geometry of a class of non-symmetric harmonic manifolds, inDifferential Geometry and Its Applications, Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992, Silesian University, Opava, 1993, pp. 415–426.Google Scholar