  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, and coordinate-free (but connection-dependent) definitions of the rough
integral of cotangent bundle-valued controlled paths, and of RDEs driven by a
rough path valued in another manifold, are given. When the path is the
realisation of semimartingale we recover the theory of It\^o integration and
SDEs on manifolds [\'E89]. We proceed to present the extrinsic counterparts to
our local formulae, and show how these extend the work in [CDL15] 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 TM, which figures in an It\^o
correction term in the parallelism RDE; such connection, which is not needed in
the geometric/Stratonovich setting, is required to satisfy properties which
guarantee well-definedness, linearity, and optionally isometricity of parallel
transport. We conclude by providing numerous examples, some accompanied by
numerical simulations, which explore the additional subtleties introduced by
our change in perspective.
