GEOMETRY OF GENERALIZED HEISENBERG GROUPS AND THEIR DAMEK-RICCI HARMONIC EXTENSIONS

It is proved that on any generalized Heisenberg group the principal cu rvatures of small geodesic spheres are invariant by local geodesic sym metries and the spectrum of the Jacobi operator is constant along geod esics. A method to compute the Jacobi fields on generalized Heisenberg groups is presented. Furthermore it is proved that a Damek-Ricci harm onic space is symmetric if and only if the spectrum of the Jacobi oper ator is constant along geodesics, or equivalently, if and only if alon g each geodesic the Jacobi operator is diagonalizable by a parallel or thonormal frame field.