In this article we consider foliations with certain properties common to several geometrically defined ones. We show that their codimension is either large or zero. As an application we obtain characterisations of the totally geodesic, complete submanifolds of S N and C pN which for most dimensions are sharper than those given by Nomizu [6], Abe [1], and the author [3]. Using this characterisation and a slightly generalized formula from [2], we prove that the standard imbedding of S" into S ~÷p is rigid for certain codimensions p, see also [7]. 2. Statement of Results Let 0(t) denote the largest integer such that the fibration