  In this paper, we analyze the geometric structure of an Euclidean submanifold
whose osculating spaces form a nonconstant family of proper subspaces of the
same dimension. We prove that if the rate of change of the osculating spaces is
small, then the submanifold must be a (submanifold of a) ruled submanifold of a
very special type. We also give a sharp estimate of the dimension of the
rulings.
