Integral manifolds of contact distributions

The existence of an integral manifold of the contact distribution (a Legendre submanifold) that passes through an arbitrary point in a contact manifold $M^{2n+1}$, in an arbitrary totally real $n$-dimensional direction is established. A Legendre submanifold with these initial data is not unique in general, but in the case of a $K$-contact manifold of dimension greater than 5 the set of these submanifolds is shown to contain a totally geodesic submanifold (which is called a Blair submanifold in the paper) if and only if this $K$-contact manifold is a Sasakian space form. Each Blair submanifold of a Sasakian space form of $\Phi$-holomorphic sectional curvature $c$ is a space of constant curvature $(c+3)/4$. Applications of these results to the geometry of principal toroidal bundles are found.