We provide a theory of manifold-valued rough paths of bounded 3 >p-variation, which we do not assume to be geometric. Rough paths are defined in charts, relying on the vector space-valued theory of [FH14FH14], and coordinate-free (but connection-dependent) definitions of the rough integral of cotangent bundle-valued controlled paths, and of rough differential equations driven by a rough path valued in another manifold, are given. When the path is the realisation of semimartingale we recover the theory of Itô integration and stochastic differential equations on manifolds [É89É89]. We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in [CDL15CDL15] to the setting of non-geometric rough paths and controlled integrands more general than 1-forms. In the
last section we turn to parallel transport and Cartan development: the lack of geometricity leads us to make the choice of a connection on the tangent bundle of the manifold T M, which figures in an Itô correction term in the parallelism rough differential equation; such connection, which is not needed inthe geometric/Stratonovich setting, is required to satisfy properties which guarantee well-definedness, linearity, and optionally isometricity of parallel transport. We conclude by providing a few examples that explore the additional subtleties introduced by our change in perspective.