Legendrian Submanifold Path Geometry

In [Ch1], Chern gives a generalization of projective geometry by considering foliations on the Grassman bundle of p-planes Gr(p, R) → R by p-dimensional submanifolds that are integrals of the canonical contact differential system. The equivalence method yields an sl(n + 1, R)valued Cartan connection whose curvature captures the geometry of such foliation. In the flat case, the space of leaves of the foliation is a second order homogeneous space [Br2]. [Ch2] deals with the geometry of the foliation of Z → Y , where Z is the bundle of Legendrian line elements over a contact threefold Y , by canonical lifts of Legendrian curves, or equivalently, the geometry of 3-parameter families of curves in the plane. An sp(2, R) -valued Cartan connection plays the role of projective connection. A generalization of [Ch2] to 4-parameter family of curves in the plane leads to a geometric realization of some exotic holonomies in dimension four [Br1]. In this paper, we generalize [Ch2] to higher dimensions. Let Z → Y 2n+1 be the bundle of Legendrian n-planes over a contact manifold Y . We consider a foliation of Z by canonical lifts of Legendrian submanifolds, which we call a Legendrian submanifold path geometry. Note that a path in this case is a Legendrian n-fold. The equivalence method provides an sp(n+1, R)-valued Cartan connection form that captures the geometry of such foliation. In the flat case, the space X of leaves of the foliation is again a second order homogeneous space. The prolonged structure equation of this second order homogeneous space is in turn that of Sp(n+1, R), which explains the appearance of a sp(n + 1, R) -valued Cartan connection form. In fact, we may consider a contact manifold Y endowed with a Legendrian submanifold path geometry structure as a union of infinitesimal homogeneous spaces