  We prove that the general fibre of the $i$-th Gauss map has dimension $m$ if
and only if at the general point the $(i+1)$-th fundamental form consists of
cones with vertex a fixed $\mathbb P^{m-1}$, 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.
