Conformal infinitesimal variations of submanifolds

Abstract: 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.