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We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way, we develop a notion of rough integration and an efficient and intrinsic theory of rough differential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold‐valued rough paths. Finally, this framework is used to show that there is a one‐to‐one correspondence between rough paths on a   d  ‐dimensional manifold and rough paths on   d  ‐dimensional Euclidean space. This last result is a rough path analogue of Cartan's development map and its stochastic version which was developed by Eells and Elworthy and Malliavin.

Constrained rough paths