# 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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by Irl Bivens
Trans. Amer. Math. Soc. 275 (1983), 397-408 Request permission

## Abstract:

Let $\mathfrak {F}$ be a smooth distribution on a Riemannian manifold $M$ with $\mathfrak {H}$ the orthogonal distribution. We say that $\mathfrak {F}$ is geodesic provided $\mathfrak {F}$ is integrable with leaves which are totally geodesic submanifolds of $M$. The notion of minimality of a submanifold of $M$ may be defined in terms of a criterion involving any orthonormal frame field tangent to the given submanifold. If this criterion is satisfied by any orthonormal frame field tangent to $\mathfrak {H}$ then we say $\mathfrak {H}$ is minimal. Suppose that $\mathfrak {F}$ and $\mathfrak {H}$ are orthogonal geodesic and minimal distributions on a submanifold of Euclidean space. Then each leaf of $\mathfrak {F}$ is also a submanifold of Euclidean space with mean curvature normal vector field $\eta$. We show that the integral of $|\eta {|^2}$ over $M$ is bounded below by an intrinsic constant and give necessary and sufficient conditions for equality to hold. We study the relationships between the geometry of $M$ and the integrability of $\mathfrak {H}$. For example, if $\mathfrak {F}$ and $\mathfrak {H}$ are orthogonal geodesic and minimal distributions on a space of nonnegative sectional curvature then $\mathfrak {H}$ is integrable iff $\mathfrak {F}$ and $\mathfrak {H}$ are parallel distributions. Similarly if ${M^n}$ has constant negative sectional curvature and dim $\mathfrak {H} = 2 < n$ then $\mathfrak {H}$ is not integrable. If $\mathfrak {F}$ is geodesic and $\mathfrak {H}$ is integrable then we characterize the local structure of the Riemannian metric in the case that the leaves of $\mathfrak {H}$ are flat submanifolds of $M$ with parallel second fundamental form.
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