Orthogonal Geodesic and Minimal Distributions

Let <!-- MATH $\mathfrak{F}$ --> be a smooth distribution on a Riemannian manifold with <!-- MATH $\mathfrak{H}$ --> the orthogonal distribution. We say that <!-- MATH $\mathfrak{F}$ --> is geodesic provided <!-- MATH $\mathfrak{F}$ --> 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 <!-- MATH $\mathfrak{H}$ --> then we say <!-- MATH $\mathfrak{H}$ --> is minimal. Suppose that <!-- MATH $\mathfrak{F}$ --> and <!-- MATH $\mathfrak{H}$ --> are orthogonal geodesic and minimal distributions on a submanifold of Euclidean space. Then each leaf of <!-- MATH $\mathfrak{F}$ --> is also a submanifold of Euclidean space with mean curvature normal vector field . We show that the integral of <!-- MATH $|\eta {|^2}$ --> over is bounded below by an intrinsic constant and give necessary and sufficient conditions for equality to hold.