Orbits of rank one and parallel mean curvature

CARLOS OLMOS

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Lugar: New York; Año: 1994 vol. 347 p. 2927 - 2939

0002-9947

Let M^n (n >= 2 ) be a (extrinsic) homogeneous irreducible fullsubmanifold of Euclidean space with rank(M) = k > 1 (i.e., it admits k > 1locally defined, linearly independent parallel normal vector fields). We provethat M must be contained in a sphere. This result toghether with previouswork of the author about homogeneous submanifolds of higher rank imply,in particular, the following theorem: A homogeneous irreducible submanifoldof Euclidean space with parallel mean curvature vector is either minimal, orminimal in a sphere, or an orbit of the isotropy representation of a simplesymmetric space.