  This paper belongs to the realm of conformal geometry and deals with
Euclidean submanifolds that admit smooth variations that are infinitesimally
conformal. Conformal variations of Euclidean submanifolds is a classical
subject in differential geometry. In fact, already in 1917 Cartan classified
parametrically the Euclidean hypersurfaces that admit nontrivial conformal
variations. Our first main result is a Fundamental theorem for conformal
infinitesimal variations. The second is a rigidity theorem for Euclidean
submanifolds that lie in low codimension.
