Isometric immersions of riemannian products

For each integer i, l<i<p, let Mt be a compact connectedriemannian manifold of dimension nt > 2, and M the riemannian product M1 x M2 x x Mp. In this paper we will prove that any codimension p isometric immersion of M in euclidean space is a product of hypersurface immersions. This means that if /: M —> E is an isometric immersion into euclidean space E of dimension N — ( Σ ί] + V > ώen there exist isometric immersions ft: Mt —• E ni+1 (1 <i < p) and a decomposition of E into a riemannian product