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Clifford systems, harmonic maps and metrics with nonnegative curvature

Chao Qian, Zizhou Tang and Wenjiao Yan

Vol. 320 (2022), No. 2, 391–424

Associated with a symmetric Clifford system {P0,P1,,Pm} on 2l, there is a canonical vector bundle η over Sl1. For m = 4 and 8, we construct explicitly its characteristic map, and determine completely when the sphere bundle S(η) associated to η 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(η) which is diffeomorphic to the inhomogeneous focal submanifold M+ of OT-FKM type isoparametric hypersurfaces with m = 3.

isoparametric hypersurface, focal submanifold, Clifford system, characteristic map, harmonic map, nonnegative sectional curvature
Mathematical Subject Classification
Primary: 55R25
Secondary: 53C20, 55Q40, 58E20
Received: 25 April 2022
Revised: 18 August 2022
Accepted: 19 August 2022
Published: 15 February 2023
Chao Qian
School of Mathematics and Statistics
Beijing Institute of Technology
Zizhou Tang
Chern Institute of Mathematics and LPMC
Nankai University
Wenjiao Yan
School of Mathematical Sciences
Beijing Normal University