Legendrian submanifold path geometry

Abstract

In [Ch1], Chern gives a generalization of projective geometry by considering foliations on the Grassmann bundle of p-planes Gr(p, T Rn+1) → Rn+1 by p-dimensional submanifolds that are integral to the canonical contact differential system. The equivalence problem gives an sl(n+2, R) valued Cartan connection whose curvature captures the geometry of the foliation. In the flat case, the space of leaves of the foliation is a second order homogeneous space [Br2]. [Ch2] deals with the foliation on the bundle Z4 → Y 3, where Z is the bundle of Legendrian line elements over a contact threefold Y , by canonical lifts of Legendrian curves, or equivalently, three parameter family of curves in the plane. An sp(2, R) valued Cartan connection plays the role of projective connection. A generalization of [Ch2] to four 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. LetZ → 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 Legendrian submanifold path geometry. Note a path in this case is a Legendrian n-fold. The equivalence problem provides an sp(n + 1, R) valued Cartan connection form that captures the geometry of such foliations. In the flat case, the space of leaves of the foliation, X, is again a second order homogeneous space. The prolonged structure equation of this flat second order homogeneous space is in turn that of Sp(n + 1, R), which explains the appearance of sp(n + 1, R) valued Cartan connection form. In fact, we may consider a contact manifold Y with Legendrian submanifold path geometry structure as a union of infinitesimal homogeneous spaces