Abstract

Associated with a symmetric Clifford system $\{P_0,P_1,\dots <!--FUNCTION\; APPLICATION-->,P_m\}$ on ${ℝ}^{2l}$, there is a canonical vector bundle $\eta$ over $S^{l-1}$. 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 $m=3$.

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