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

On some aspects of the geometry of non integrable distributions and applications