Orthogonal geodesic and minimal distributions

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.