It is then natural to try to understand all submanifolds for which equality in (1) holds everywhere. In euclidean space, Chen showed that the trivial examples satisfying his basic equality are either affine subspaces or rotation hypersurfaces obtained by rotating a straight line, that is, cones and cylinders. Nontrivial examples for n ≥ 4 divide in two classes, namely, any minimal submanifolds of rank two, which we completely describe in [DF], and a certain class of nonminimal submanifolds foliated by totally umbilic spheres of codimension two.
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