| We consider a generalization of path geometry to a contact manifold, i.e., the geometric structure on a contact manifold Y2n+1 that models the second order homogeneous space Spn+1,R→RP2n +1, called Legendrian submanifold path geometry. Such a structure on Y associates to each Legendrian n-plane E ⊂ TY a unique Legendrian submanifold of Y tangent to E. We show that given such a geometry there exists an sp (n + 1, R)-valued Cartan connection form, called a symplectic connection, that is canonically defined via the method of equivalence.;A special class of these geometries is characterized by having a well defined conformal class of symmetric (n + 1) differentials on the space of Legendrian submanifold paths X. The vanishing of this symmetric differential represents a necessary condition for the contact of neighboring Legendrian leaves. A double fibration naturally arises, and we give a dual interpretation of the contact manifold Y in terms of X. The G structure induced on X provides an example of a classical, non-metric, irreducible holonomy GL(n + 1, R) as represented on sym2(Rn+1). |
