A class of minimal submanifolds in spheres

Abstract

We introduce a class of minimal submanifolds Mⁿ, n ≥ 3, in spheres 𝕊ⁿ⁺² that are ruled by totally geodesic spheres of dimension n − 2. If simply-connected, such a submanifold admits a one-parameter associated family of equally ruled minimal isometric deformations that are genuine. As for compact examples, there are plenty of them but only for dimensions n = 3 and n = 4. In the first case, we have that M³ must be a 𝕊¹-bundle over a minimal torus T² in 𝕊⁵ and in the second case M⁴ has to be a 𝕊²-bundle over a minimal sphere 𝕊² in 𝕊⁶. In addition, we provide new examples in relation to the well-known Chern-do Carmo–Kobayashi problem since taking the torus T² to be flat yields minimal submanifolds M³ in 𝕊⁵ with constant scalar curvature.