Clifford systems, Cartan hypersurfaces and Riemannian submersions

Abstract

Cartan hypersurfaces are minimal isoparametric hypersurfaces with 3 distinct constant principal curvatures in unit spheres. In this article, we firstly build a relationship between the focal submanifolds of Cartan hypersurfaces and the Hopf fiberations and give a new proof of the classification result on Cartan hypersurfaces. Nextly, we show that there exists a Riemannian submersion with totally geodesic fibers from each Cartan hypersurface M3m to the projective planes \({{\mathbb{F}}P^2}\) (\({{\mathbb{F}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}},{\mathbb{O}}}\) for m = 1, 2, 4, 8, respectively) endowed with the canonical metrics. As an application, we give several interesting examples of Riemannian submersions satisfying a basic equality due to Chen (Proc Jpn Acad Ser A Math Sci 81:162–167, 2005).