Clifford systems, harmonic maps and metrics with nonnegative curvature

Chao Qian, Zizhou Tang and Wenjiao Yan

Vol. 320 (2022), No. 2, 391–424

DOI: 10.2140/pjm.2022.320.391

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 $m=3$.

Keywords

isoparametric hypersurface, focal submanifold, Clifford system, characteristic map, harmonic map, nonnegative sectional curvature

Mathematical Subject Classification

Primary: 55R25

Secondary: 53C20, 55Q40, 58E20

Milestones

Received: 25 April 2022

Revised: 18 August 2022

Accepted: 19 August 2022

Published: 15 February 2023