Legendrian Cone Structures and Contact Prolongations

Abstract

We exploit a natural correspondence between holomorphic (2, 3, 5)-distributions and nondegenerate lines on holomorphic contact manifolds of dimension 5 to present a new perspective in the study of symmetries of (2, 3, 5)-distributions. This leads to a number of new results in this classical subject, including an unexpected relation between the multiply-transitive families of models having 7-and 6-dimensional symmetries, and a one-to-one correspondence between equivalence classes of nontransitive (2, 3, 5)-distributions with 6-dimensional symmetries and nonhomogeneous nondegenerate Legendrian curves in P 3 . An ingredient for establishing the former is an explicit classification of homogeneous nondegenerate Legendrian curves in P 3 , which we present.

Minimal rational curves and 1-flat irreducible G-structures

“…They are also properties of projective Legendrian submanifolds with nondegenerate second fundamental forms and can be deduced from pp. 348-349 of [22] (see also Proposition 2 in [13]). ( 5) can be checked by elementary arguments for the case (ii) of Proposition 2.5.…”