Totally umbilical manifolds inG(2,n). I

Abstract

It is well known that every totally umbilical submanifold of a space of constant curvature is either a small sphere or is totally geodesic. B.-Y. Chen has classified totally umbilical submanifolds of compact, rank one, symmetric spaces ([4], [5]): in particular, they are all extrinsic spheres, that is, they have a parallel mean curvature vector H (or are totally geodesic). In this paper totally umbilical submanifolds Fl of dimension l ≥ 3 are classified in the irreducible symmetric space that is "next in complexity": Grassmann manifold G(2, n). Such submanifolds are either 1) totally geodesic [3] or 2) extrinsic spheres [small spheres in totally geodesic spheres; their position in G(2, n) is described here] or 3) essentially totally umbilical (H ≠ 0, ∇ ⊥ H ≠ 0). If the submanifold is of type 3), then it is either a) an umbilical hypersurface of nonconstant mean curvature in totally geodesic S1 × S1 ⊂ G(2, n) or b) an "oblique diagonal," a diagonal of the product of two small spheres of different radii in totally geodesic Sl+1 × Sl+1 ⊂ G(2, n) (it has constant mean and sectional curvatures). Submanifolds 3a) and 3b) are described completely. The latter of the two negates two of Chen's conjectures. It is shown that submanifold Fl ⊂ El+2 (l ≥ 3) with a totally umbilical Grassmannian image has a totally geodesic Grassmannian image and is classifiable [11].