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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.

MSC:
53B20 Local Riemannian geometry
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