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A branched rough path $X$ consists of a rough integral calculus for $X \colon[0, T] \to \mathbb R^d$ which may fail to satisfy integration by parts. UsingKelly's bracket extension [Kel12], we define a notion of pushforward ofbranched rough paths through smooth maps, which leads naturally to a definitionof branched rough path on a smooth manifold. Once a covariant derivative isfixed, we are able to give a canonical, coordinate-free definition of integralagainst such rough paths. After characterising quasi-geometric rough paths interms of their bracket extension, we use the same framework to definemanifold-valued rough differential equations (RDEs) driven by quasi-geometricrough paths. These results extend previous work on $3 > p$-rough paths[ABCRF22], itself a generalisation of the Ito calculus on manifolds developedby Meyer and Emery [Mey81, E89, E90], to the setting of non-geometric roughcalculus of arbitrarily low regularity.

A Transfer Principle for Branched Rough Paths