Michel Rumin is a student of Mikhael Gromov, who asked him the following question : Let M be a manifold with contact structure ξ, E a vector bundle over M. A partial connection on E is a covariant derivative ∇ve defined for smooth sections e of E but only for vectors v in ξ. In particular, parallel translation is defined only along Legendrian curves, that is curves which are tangent to ξ. Can one define the curvature of such a connection?

Gromov provided the following hint : For an ordinary connection A, curvature arises in the asymptotics of holonomy around short loops. A loop encompasses a certain “span” (a 2-vector, see below), quadratic in length, and holonomy deviates from the identity by an amount proportional to curvature times span, that is, quadratic in length. In case M has dimension 3 and carries a contact structure, then every Legendrian loop has essentially zero area. Gromov conjectured that, in this case, curvature should arise as the cubic term in the asymptotic expansion of holonomy.

Michel Rumin has found a notion of curvature for partially defined connections along the above lines. The point is to understand the exterior differential for a partially defined 1-form. In fact, M. Rumin constructs a substitute for the de Rham complex : a locally exact complex of hypoelliptic operators naturally attached to a contact manifold (M, ξ) of dimension 2m+ 1. The operator which sends m-forms to m+1-forms is new. It is of second order.