  We consider a regular distribution $\mathcal{D}$ in a Riemannian manifold
$(M,g)$. The Levi-Civita connection on $(M,g)$ together with the orthogonal
projection allow to endow the space of sections of $\mathcal{D}$ with a natural
covariant derivative, the intrinsic connection. Hence we have two different
covariant derivatives for sections of $\mathcal{D}$, one directly with the
connection in $(M,g)$ and the other one with this intrinsic connection. Their
difference is the second fundamental form of $\mathcal{D}$ and we prove it is a
significant tool to characterize the involutive and the totally geodesic
distributions and to give a natural formulation of the equation of motion for
mechanical systems with constraints. The two connections also give two
different notions of curvature, curvature tensors and sectional curvatures,
which are compared in this paper with the use of the second fundamental form.
