Controlled Rough Paths on Manifolds

We build the foundation for a theory of controlled rough pathson manifolds. A number of natural candidates for the definition of manifoldvalued controlled rough paths are developed and shown to be equivalent. Thetheory of controlled rough one-forms along such a controlled path and theirresulting integrals are then defined. This general integration theory doesrequire the introduction of an additional geometric structure on the manifoldwhich we refer to as a "parallelism." Thetransformation properties of the theory under change of parallelisms isexplored. Using these transformation properties, it is shown that theintegration of a smooth one-form along a manifold valued controlled rough pathis in fact well defined independent of any additional geometric structures. Wepresent a theory of push-forwards and show how it is compatible with ourintegration theory. We give a number of characterizations for solvinga rough differential equation when the solution is interpreted as a controlledrough path on a manifold and then show such solutions exist and are unique. We develop the notion of parallel translation along a controlled rough path. This lends itself to a theory of rolling and unrolling maps for not only controlled rough paths but controlled rough one-forms