Compositions of isometric immersions in higher codimension

Given a submanifold M-n of Euclidean space Rn+p with codimension p less tha n or equal to 6, under generic conditions on its second fundamental form, w e show that any other isometric immersion of M-n into Rn+p+q, 0 less than o r equal to q : n - 2p - 1 and 2q less than or equal to n + 1 if q greater t han or equal to 5, must be locally a composition of isometric immersions. T his generalizes several previous results on rigidity and compositions of su bmanifolds. We also provide conditions under which our result is global.