zbMATH — the first resource for mathematics

Manifolds whose curvature operator has constant eigenvalues at the basepoint. (English) Zbl 0797.53010
Let \(M\) be a Riemannian manifold. Osserman conjectured that if the Jacobi operator \({\mathcal R}(X): Z \to R(X,Z)X\) has constant eigenvalues, then \(M\) is locally a rank 1 symmetric space or is flat. The pointwise question is considerably more complicated. Examples of Riemannian manifolds are constructed so \(\mathcal R\) has constant eigenvalues at the basepoint, but \(\mathcal R\) is not the curvature operator of a rank 1 symmetric space and is not trivial; this construction uses Clifford modules.

53B20 Local Riemannian geometry
Full Text: DOI
[1] Adams, J. Vector fields on spheres.Annals of Math. 75, 603–632 (1962). · Zbl 0112.38102
[2] Atiyah, M. F., Bott, R., and Shapiro, A. Clifford modules.Topology 3 (suppl. 1), 3–38 (1964). · Zbl 0146.19001
[3] Chi, Q-S. A curvature characterization of certain locally rank-1 symmetric spaces.J. Diff. Geom. 28, 187–202 (1988). · Zbl 0654.53053
[4] Osserman, R. Curvature in the 80’s.Am. Math. Monthly, 731–756 (1990). · Zbl 0722.53001
[5] Steenrod, N.Topology of Fiber Bundles. Princeton, Princeton, NJ, 1965. · Zbl 0132.19201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.