We define higher order fundamental forms and osculating spaces of projective algebraic varieties, using sheaves of principal parts. We show that the mth fundamental form can be viewed as the differential of the (m−1)th Gauss map, and explain why the vanishing of the mth fundamental form implies that the variety is contained in a general (m − 1)th osculating space. Pointwise, the fundamental forms give linear systems on the projectivized tangent spaces. We show that, at each point, the Jacobian of the mth fundamental form is contained in the (m − 1)th fundamental form. In the case of ruled varieties, we describe these linear systems. We discuss conditions for a surface to be ruled, in terms of the second fundamental form and the Fubini cubic.
