Abstract
Here we study three examples of differential geometry behaviour of projective varieties: (1) the classification of smooth surfaces in P^{2n+1} whose m-osculating space have everywhere dimension 2m (1<=m<=n); (2) hyper-surfaces with Hessian rank 0; (3) singular hyper-surfaces in weighted projective spaces whose tangent sheaf is locally free and a sub-bundel of the restricted tangent bundle