D’Atri Spaces of Type Emphasis Type="Italic"k/E

Authors: Teresa AriasMarco Maria J Druetta

Publish Date: 2012/10/03

Volume: 24, Issue: 2, Pages: 721-739

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Abstract

In this article we continue the study of the geometry of kD’Atri spaces 1≤k ≤n−1 n denotes the dimension of the manifold begun by the second author It is known that kD’Atri spaces k≥1 are related to properties of Jacobi operators R v along geodesics since she has shown that operatornametrR v operatornametrR v2 are invariant under the geodesic flow for any unit tangent vector v Here assuming that the Riemannian manifold is a D’Atri space we prove in our main result that operatornametrR v3 is also invariant under the geodesic flow if k≥3 In addition other properties of Jacobi operators related to the Ledger conditions are obtained and they are used to give applications to Iwasawa type spaces In the class of D’Atri spaces of Iwasawa type we show two different characterizations of the symmetric spaces of noncompact type they are exactly the frakCspaces and on the other hand they are k D’Atri spaces for some k≥3 In the last case they are kD’Atri for all k=1…n−1 as well In particular Damek–Ricci spaces that are k D’Atri for some k≥3 are symmetricFinally we characterize kD’Atri spaces for all k=1…n−1 as the frakSCspaces geodesic symmetries preserve the principal curvatures of small geodesic spheres Moreover applying this result in the case of 4 dimensional homogeneous spaces we prove that the properties of being a D’Atri 1D’Atri space or a 3D’Atri space are equivalent to the property of being a kD’Atri space for all k=123The authors were partially supported by CONICET FONCyT and SECyT UNC The first author’s work has also been supported by DGI Spain and FEDER Project MTM201015444 by Junta de Extremadura and FEDER funds by DFG Sonderforschungsbereich 647 and the program “Becas Iberoamérica Jóvenes Profesores e Investigadores Santander Universidades” of Santander Bank

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