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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Orbits of rank one and parallel mean curvature


Author: Carlos Olmos
Journal: Trans. Amer. Math. Soc. 347 (1995), 2927-2939
MSC: Primary 53C30; Secondary 53C42
MathSciNet review: 1308018
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1308018-4
PII: S 0002-9947(1995)1308018-4
Article copyright: © Copyright 1995 American Mathematical Society