Abstract
LetR k be the higher-order curvature operator of a Riemannian manifoldM. IfR k has constant eigenvalues at the basepointP ofM for all oddk, then all the odd covariant derivatives of the curvature tensor vanish atP. If the metric is real analytic, the geodesic involution is a local isometry atP.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Besse, A. L. Manifolds All of Whose Geodesics are Closed. New York: Springer-Verlag 1978.
Chi, Q.-S. A curvature characterization of certain locally rank-1 symmetric spaces. J. Diff. Geom.28, 187–202 (1988).
Gilkey, P. Manifolds whose curvature operator has constant eigenvalues at the basepoint. Preprint.
Helgason, S. Differential Geometry and Symmetric Spaces. New York: Academic Press 1962.
Osserman, R. Curvature in the 80’s. Am. Math. Monthly97, 731–756 (1990).
Szabó, Z. I. A simple topological proof for the symmetry of 2 point homogeneous spaces. Invent. Math.106, 61–64 (1991).
Vanhecke, L., and Willmore, T. J. Interaction of spheres and tubes. Math. Ann.263, 31–42 (1983).
Author information
Authors and Affiliations
Additional information
Research partially supported by the NSF and IHES.
Rights and permissions
About this article
Cite this article
Gilkey, P.B. Manifolds whose higher odd order curvature operators have constant eigenvalues at the basepoint. J Geom Anal 2, 151–156 (1992). https://doi.org/10.1007/BF02921386
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02921386