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A curvature characterization of certain locally rank-one symmetric spaces. (English) Zbl 0654.53053
Let R(X,Y)Z denote the curvature tensor of a Riemannian manifold M. Define the Jacobi operator \(K_ v(.):=R(.,v)v\) for each unit tangent vector v. Suppose \(K_ v\) has constant eigenvalues (counting multiplicities) independent of v. It is proved that in this case 1. if M is odd dimensional, then M is of constant sectional curvature, 2. if dim M\(=2(2k+1)\), \(k\geq 0\), or dim M\(=4\), then M has either constant curvature or is covered by a standard complex projective space or its noncompact dual, 3. if M is nonflat and Kählerian, then M has constant holomorphic sectional curvature if the sectional curvatures are all nonpositive or all nonnegative.
Reviewer: V.Zoller

MSC:
53C35 Differential geometry of symmetric spaces
53C20 Global Riemannian geometry, including pinching
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