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