Orbits of rank one and parallel mean curvature

Olmos, Carlos

doi:10.1090/S0002-9947-1995-1308018-4

Orbits of rank one and parallel mean curvature
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by Carlos Olmos

Trans. Amer. Math. Soc. 347 (1995), 2927-2939

DOI: https://doi.org/10.1090/S0002-9947-1995-1308018-4

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Abstract:

Let ${M^n}(n \geqslant 2)$ be a (extrinsic) homogeneous irreducible full submanifold of Euclidean space with $rank(M) = k \geqslant 1$ (i.e., it admits $k \geqslant 1$ locally defined, linearly independent parallel normal vector fields). We prove that $M$ must be contained in a sphere. This result toghether with previous work of the author about homogeneous submanifolds of higher rank imply, in particular, the following theorem: A homogeneous irreducible submanifold of Euclidean space with parallel mean curvature vector is either minimal, or minimal in a sphere, or an orbit of the isotropy representation of a simple symmetric space.

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