On higher Gauss maps

We prove that the general fibre of the i-th Gauss map has dimension m   if and only if at the general point the (i+1)(i+1)-th fundamental form consists of cones with vertex a fixed Pm−1Pm−1, extending a known theorem for the usual Gauss map. We prove this via a recursive formula for expressing higher fundamental forms. We also show some consequences of these results.