A class of minimal submanifolds in spheres

摘要

We introduce a class of minimal submanifolds M~n, n ≥ 3, in spheres S~(n+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 M~3 must be a S~1-bundle over a minimal torus T~2 in S~5 and in the second case M~4 has to be a S~2-bundle over a minimal sphere S~2 in S~6. 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 M~3 in S~5 with constant scalar curvature.