  In this paper we study geometric aspects of dynamics generated by Young
differential equations (YDE) driven by $\alpha$-H\"older trajectories with
$\alpha \in (1/2, 1)$. We present a number of properties and geometrical
constructions on this low regularity context: Young It\^o geometrical formula,
horizontal lift in principal fibre bundles, parallel transport, covariant
derivative, development and anti-development, among others. Our main
application here is a geometrical decomposition of flows generated by YDEs
according to diffeomorphisms generated by complementary distributions
(integrable or not). The proof of existence of this decomposition is based on
an Young It\^o-Kunita formula for $\alpha$-H{\"o}lder paths proved by
Castrequini and Catuogno (Chaos Solitons Fractals, 2022).
