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Transactions of the American Mathematical Society

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

 

 

Orthogonal geodesic and minimal distributions


Author: Irl Bivens
Journal: Trans. Amer. Math. Soc. 275 (1983), 397-408
MSC: Primary 53C12; Secondary 57R30
MathSciNet review: 678359
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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 $ \vert\eta {\vert^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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DOI: https://doi.org/10.1090/S0002-9947-1983-0678359-4
Keywords: Geodesic distribution, minimal distribution, parallel distribution, foliation, minimal submanifold, Codazzi tensor
Article copyright: © Copyright 1983 American Mathematical Society