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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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Codazzi tensors and reducible submanifolds
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by Irl Bivens PDF
Trans. Amer. Math. Soc. 268 (1981), 231-246 Request permission

Abstract:

An integral formula is derived for Codazzi tensors of type $(k, k)$. Many of the classical Minkowski type integral formulas then become special cases of this one. If $M$ is a submanifold of Euclidean space and $\pi$ is a parallel distribution on $M$ then each leaf of $\pi$ is a submanifold of Euclidean space with mean curvature normal vector field $\eta$. Using the above integral formula we show that the integral of ${\left | \eta \right |^2}$ over $M$ is bounded below by an intrinsic constant and we give necessary and sufficient conditions for equality to hold. The reducible surfaces for which equality holds are characterized and related results concerned with Riemannian product manifolds are proved. Parallel tensors of type $(1, 1)$ are characterized in terms of the de Rham decomposition. It is shown that if $M$ is irreducible and $A$ is a parallel tensor of type $(1, 1)$ on $M$ which is not multiplication by a constant then $M$ is a Kaehler manifold. Some further results are derived for manifolds whose simply connected cover is Kaehler.
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Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 268 (1981), 231-246
  • MSC: Primary 53C40
  • DOI: https://doi.org/10.1090/S0002-9947-1981-0628456-2
  • MathSciNet review: 628456