Authors’ abstract: “Given an Itô vector field \(M\), there is a unique solution \(\xi^t(h)\) to the differential equation \[ {d\xi^t(h) \over dt}= M\bigl(\xi^t(h)\bigr), \quad \xi^0 (h)=h \] for any continuous and piecewise smooth path \(h\). We show that for any \(t\in \mathbb{R}\), the map \(h\to \xi^t(h)\) is continuous in the \(p\)-variation topology for any \(p\geq 1\), so that it uniquely extends to a solution flow on the space of all geometric rough paths. Applying this result to B. K. Driver’s geometric flow equation on the path space over a closed Riemannian manifold \[ {d\zeta^t \over dt} =X^h (\zeta^t), \quad \xi^0= \text{id}, \] where \(X^h\) is the vector field defined by parallel translating \(h\) via a connection, our result especially yields a deterministic construction of Driver’s flow”.