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 be a smooth distribution on a Riemannian manifold with the orthogonal distribution. We say that is geodesic provided is integrable with leaves which are totally geodesic submanifolds of . The notion of minimality of a submanifold of 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 then we say is minimal. Suppose that and are orthogonal geodesic and minimal distributions on a submanifold of Euclidean space. Then each leaf of is also a submanifold of Euclidean space with mean curvature normal vector field . We show that the integral of over 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 and the integrability of . For example, if and are orthogonal geodesic and minimal distributions on a space of nonnegative sectional curvature then is integrable iff and are parallel distributions. Similarly if has constant negative sectional curvature and dim then is not integrable. If is geodesic and is integrable then we characterize the local structure of the Riemannian metric in the case that the leaves of are flat submanifolds of with parallel second fundamental form.

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Additional Information

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