A "geometric" rough path is a rough path such that $Sym(\mathbb{X}\_{s,t})=\frac{1}{2}X\_{s,t}\otimes X\_{s,t}$. For example the Ito rough path is not geometric because $Sym(\mathbb{X}\_{s,t})=\frac{1}{2}X\_{s,t}\otimes X\_{s,t}-\frac{1}{2}I(t-s)$ but the Stratonovich rough path is geometric.


Why the term "geometric" here? Is there some intuition I'm missing? When I think geometric, I think of geometric Brownian motion, (<http://en.wikipedia.org/wiki/Geometric_Brownian_motion>) or geometric series, something with an exponent. I don't see that here.



Why the term "geometric" rough path?
Geometric rough paths have the property that if you want to solve an equation with values in a manifold, choose a coordinate chart, and write in local coordinates
$$
dY^i = V\_0^i(Y)\,dt + \sum\_j V\_j^i(Y)\,dX\_j
$$
for some vector fields $V\_i$ (with the obvious abuse of notation that the solution actually depends on the choice of $\mathbb{X}$, not just on $X$), then the solution does not depend on the choice of chart. I believe that this is the reason why this terminology was chosen.


In other words, solutions to equations driven by "geometric" rough paths transform according to the usual chain rule, rather than some version of Itô's formula. This is also why people studying SDEs on manifolds tend to write everything in Stratonovich form, rather than in Itô form.


