CLIFFORD SYSTEMS, HARMONIC MAPS AND METRICS WITH NONNEGATIVE CURVATURE

Associated with a symmetric Clifford system {P-0, P-1,..., P-m} on R-2l, there is a canonical vector bundle eta over Sl-1. For m = 4 and 8, we construct explicitly its characteristic map, and determine completely when the sphere bundle S(eta) associated to eta admits a cross-section. These generalize the results of Steenrod (1951) and James (1958). As an application, we establish new harmonic representatives of certain elements in homotopy groups of spheres (see [Peng and Tang 1997; 1998]). By a suitable choice of Clifford system, we construct a metric of nonnegative curvature on S(eta) which is diffeomorphic to the inhomogeneous focal submanifold M+ of OT-FKM type isoparametric hypersurfaces with m = 3.