Let \(S=(M,r)\) be a submanifold of a Riemannian manifold \(\overline M\), where \(r\) is an immersion of \(M\) into \(\overline M\); let \(\nu:I\times M\to\overline M\) be a deformation of \(S\) and \(Z\) the deformation field of \(\nu\). Then \(Z\) is normal if the tangential component of \(Z\) is everywhere zero. A deformation \(\nu\) is an infinitesimal isometric deformation (I.I.D) if and only if \(g(\overline D_ xZ,Y)+g(X,\overline D_ yZ)=0\), \(X,Y\in{\mathfrak X}(M)\) where \(\overline D\) is the covariant differentiation operator of the Riemannian manifold \((\overline M,g)\). If \(Z\) is a normal and I.I.D. of \(S\), then \(S\) is totally geodesic whenever \(Z\) is nonzero. In this paper the above properties are generalized for the Cartesian product of Riemannian manifolds. Some interesting results concerning the geometry of products of manifolds are studied.