Orthogonal geodesic and minimal distributions

Abstract

Let F \mathfrak {F} be a smooth distribution on a Riemannian manifold M M with H \mathfrak {H} the orthogonal distribution. We say that F \mathfrak {F} is geodesic provided F \mathfrak {F} is integrable with leaves which are totally geodesic submanifolds of M M . The notion of minimality of a submanifold of M 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 H \mathfrak {H} then we say H \mathfrak {H} is minimal. Suppose that F \mathfrak {F} and H \mathfrak {H} are orthogonal geodesic and minimal distributions on a submanifold of Euclidean space. Then each leaf of F \mathfrak {F} is also a submanifold of Euclidean space with mean curvature normal vector field η \eta . We show that the integral of | η | 2 |\eta {|^2} over M 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 M and the integrability of H \mathfrak {H} . For example, if F \mathfrak {F} and H \mathfrak {H} are orthogonal geodesic and minimal distributions on a space of nonnegative sectional curvature then H \mathfrak {H} is integrable iff F \mathfrak {F} and H \mathfrak {H} are parallel distributions. Similarly if M n {M^n} has constant negative sectional curvature and dim H = 2 > n \mathfrak {H} = 2 > n then H \mathfrak {H} is not integrable. If F \mathfrak {F} is geodesic and H \mathfrak {H} is integrable then we characterize the local structure of the Riemannian metric in the case that the leaves of H \mathfrak {H} are flat submanifolds of M M with parallel second fundamental form.