Clifford systems, harmonic maps and metrics with nonnegative curvature (Q2685168)

This paper is a careful study of certain vector bundles over spheres, defined by Clifford systems \[ \{P_0, P_1, \dots P_m\} \text{ on } \mathbb{R}^{2 \ell}. \] The bulk of the paper is devoted to proving Theorem 1.1, which gives explicit formulae for the clutching (`characteristic') maps of these bundles when \(m=4,8\). This includes the classical special case of the Stiefel manifold \(V_2(\mathbb{H}^k)\) when \(m=4\), and includes \(V_2(\mathbb{O}^k)\) when \(m=8\). As a consequence, the authors determine precisely when the associated sphere bundle admits a cross-section (Propositions 1.2 and 1.3). For the classical special case \(V_2(\mathbb{H}^k)\) this result is due to James, who asserted the result for \(V_2(\mathbb{O}^k)\) without proof. The paper concludes with two shorter sections. The first combines the preceding results with work of \textit{W.-Y. Ding} [Int. J. Math. 5, No. 6, 849--860 (1994; Zbl 0822.58011)] to show that a certain family of classes in the homotopy groups of spheres admit harmonic representatives (Theorem 1.5). The second section exhibits a Clifford system with \(m=3\) for which the so-called focal submanifold \(M_{+}\) has nonnegative sectional curvature (Theorem 1.7).