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Annali di Matematica Pura ed Applicata

, Volume 191, Issue 4, pp 677–709 | Cite as

Conformally Osserman manifolds of dimension 16 and a Weyl–Schouten theorem for rank-one symmetric spaces

  • Y. Nikolayevsky
Article

Abstract

A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. The Osserman Conjecture asserts that any Osserman manifold is either flat or rank-one symmetric. We prove that both the Osserman Conjecture and its conformal version, the Conformal Osserman Conjecture, are true, modulo a certain assumption on algebraic curvature tensors in \({\mathbb {R}^{16}}\). As a consequence, we show that a Riemannian manifold having the same Weyl tensor as a rank-one symmetric space is conformally equivalent to it.

Keywords

Osserman manifold Jacobi operator Clifford structure 

Mathematics Subject Classification (2000)

53B20 53A30 

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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLa Trobe UniversityMelbourneAustralia

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