Bruce Driver University of California, San Diego (UCSD) In this talk I will describe recent joint work with Thomas Cass and Christian Litterer on rough paths which are ``constrained’’ to lie in a d - dimensional submanifold of a Euclidean space E. We will begin by defining this notion and then proceed to describe the (second) order geometric calculus which arises out of this theory. The talk will conclude with a rough version of Cartan’s development map which parameterizes all constrained rough paths by rough paths in a d - dimensional Euclidean space.