Infinitesimal variations of submanifolds

This paper deals with the subject of infinitesimal variations Euclidean submanifolds arbitrary dimension and codimension. The main goal is to establish a Fundamental theorem for these geometric objects. Similar theory isometric immersions in space, we prove that system three equations certain pair tensors are integrability conditions differential equation determines variations. In addition, give some rigidity results when submanifold intrinsically Riemannian product manifolds.