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.
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