On certain locally homogeneous Clifford manifolds

Abstract

Given a manifoldM, a Clifford structure of orderm onM is a family ofm anticommuting complex structures generating a subalgebra of dimension 2 m of End(T(M)). In this paper we investigate the existence of locally invariant Clifford structures of orderm≥2 on a class of locally homogeneous manifolds. We study the case of solvable extensions ofH-type groups, showing in particular that the solvable Lie groups corresponding to the symmetric spaces of negative curvature carry invariant Clifford structures of orderm≥2. We also show that for eachm and any finite groupF, there is a compact flat manifold with holonomy groupF and carrying a Clifford structure of orderm.