Reducibility of isometric immersions

Abstract:

For $i = 1,2$, suppose that the connected riemannian manifold ${M_i}$ possesses a codimension ${p_i}$ euclidean isometric immersion whose first normal space has dimension ${p_i}$ and whose type number is at least two at each point, and let $N = \dim ({M_1} \times {M_2}) + {p_1} + {p_2}$. In this note it is proven that if f is any isometric immersion from the riemannian product ${M_1} \times {M_2}$ into euclidean N-space ${E^N}$, then there exists an orthogonal decomposition ${E^N} = {E^{{N_1}}} \times {E^{{N_2}}}$ together with isometric immersions ${f_i}:{M_i} \to {E^{{N_i}}}$ such that $f = {f_1} \times {f_2}$.