In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form $$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$ where $X$ is a semimartingale on a manifold $M$ and $\Theta$ is a predictable process above $X$ taking values in the second order tangent bundle $\tau M$ (the bundle of second order differential operators without constant term). See also "An invitation to second order stochastic differential geometry".
In the above, $\mathbf{d}X_t$ is the second order vector $$\mathbf{d}X_t = dX^i_t D_i +\frac{1}{2}d [X^i_t, X^j_t] D_{ij},$$ the existence of which is "only metaphysical" (to speak with Emery's words) and the actual definition of the integral above does not use this "identity"; instead it is defined axiomatically.
Now I wonder: What is the relation of this to the theory of rough paths? In particular, can one make rigorous sense of $\mathbf{d}X_t$ as a covariant object using the theory of rough paths on a manifold, i.e. for a rough path, is its (second order?) infinitesimal increment somehow an element of the second order tangent bundle at a point? Can one then make sense pathwise of the integral above?
And what is actually a good covariant definition of a rough path on a manifold in this setting? I must admit that I have a hard time understanding rough paths in a covariant setting.
