Isoparametric foliations and complex structures

Explicit representations of complex structures on closed manifolds are valuable, but relatively rare in the literature. By using isoparametric theory, we construct complex structures on isoparametric hypersurfaces with [Formula: see text], [Formula: see text] in the unit sphere, as well as on focal submanifolds [Formula: see text] of OT-FKM type with [Formula: see text], [Formula: see text] and [Formula: see text] in the definite case. Furthermore, we investigate the existence and non-existence of invariant (almost) complex structures on homogeneous isoparametric hypersurfaces with [Formula: see text], providing a complete classification of those that admit such structures. Finally, we discuss the geometric properties of the complex structures arising from isoparametric theory.