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.