Geometry of D'Atri spaces of type k

MARIA J. DRUETTA

ANNALS OF GLOBAL ANALYSIS AND GEOMETRY

pre-print, enviado

Lugar: Kluwer Academic; enviado a publicar, octubre de 2009; Año: 2009 vol. 1 p. 1 - 20

0232-704X

 A Riemannian n-dimensional manifold M is a D´Atri space of type k (or k-D´Atri space), k=1,...,n-1, if the geodesic symmetries preserve the k-th  elementary symmetric functions of the eigenvalues of the shape operators of all small geodesic spheres in M. Symmetric spaces are k-D´Atri spaces for all possible k=1,...,n-1 and the property 1-D´Atri is the D´Atri condition in  the usual sense.  In this article we study some aspects of the geometry of k-D´Atri spaces, in particular those related to properties of Jacobi operators along geodesics. We show that k-D´Atri spaces for all k=1,...,l satisfy that the traze of the the k-th power of  R(.,v)v, v a unit vector in TM, is invariant under the geodesic flow for all k=1,...,l. Further, if M is k-D´Atri for all k=1,...,n-1, then the eigenvalues of Jacobi operators are constant functions along geodesics.  In the case of spaces of Iwasawa type, we show that k-D´Atri spaces for all k=1,...,n-1 are exactly the symmetrics spaces of noncompact type. Moreover, in the class of Damek-Ricci spaces, the symmetric spaces of rank one are characterized as those that are 3-D´Atri.