On varieties with higher osculating defect
  In this paper, using the method of moving frames, we generalise some of Terracini's results on varieties with tangent defect. In particular, we characterise varieties with higher order osculating defect in terms of Jacobians of higher fundamental forms and moreover we characterise varieties with "small" higher fundamental forms as contained in scrolls. 
Pietro De Poi, Roberta Di Gennaro and Giovanna Ilardi
math.AG math.DG