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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Orbits of rank one and parallel mean curvature
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by Carlos Olmos PDF
Trans. Amer. Math. Soc. 347 (1995), 2927-2939 Request permission

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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 2927-2939
  • MSC: Primary 53C30; Secondary 53C42
  • DOI: https://doi.org/10.1090/S0002-9947-1995-1308018-4
  • MathSciNet review: 1308018