Homogeneous submanifolds of higher rank and parallel mean curvature (Q1330394)

The purpose of this article is to investigate the following problem: When is a homogeneous submanifold of a Euclidean space an orbit of the isotropy representation of a simple symmetric space? The author proves the following results in this respect. Theorem 1. If \(M^ n\), \(n \geq 2\), is a compact homogeneous irreducible full submanifold of the Euclidean space with rank \((M^ n) \geq 2\), then \(M^ n\) is an orbit of the isotropy representation of a simple symmetric space. Theorem 2. If \(M^ n\) is a compact homogeneous irreducible submanifold of the Euclidean space with parallel mean curvature vector which is not minimal in a sphere, then \(M^ n\) is an orbit of the isotropy representation of a simple symmetric space. By the rank of a submanifold the author means the maximal number of linearly independent (locally defined) parallel normal vector fields.