A note on the duality principle and Osserman condition

AbstractIn this note we prove that for a Riemannian manifold the Osserman point-wise condition is equivalent to the Raki´c duality principle. 1 Introduction Let R be an algebraic curvature tensor on a Euclidean space R n and let forX ∈ R n , R X : Y → R(Y,X)X be the corresponding Jacobi operator. Analgebraic curvature tensor R is called Osserman , if the spectrum of the Jacobioperator R X does not depend on the choice of a unit vector X ∈ R n .Let M n be a Riemannian manifold, R its curvature tensor and R X the cor-responding Jacobi operator. It is well known that the properties of R X areintimately related with the underlying geometry of the manifold. The manifoldM n is called pointwise Osserman if R is Osserman at every point p ∈ M n , andis called globally Osserman if the spectrum of R X is the same for all X in theunit tangent bundle of M n . Locally two-point homogeneous spaces are globallyOsserman, since the isometry group of each of these spaces is transitive on itsunit tangent bundle. Osserman [O] conjectured that the converse is also true.This gives a very nice characterisation of local two-point homogeneous spacesin terms of the geometry of the Jacobi operator.