  We give a generalized curvature-dimension inequality connecting the geometry
of sub-Riemannian manifolds with the properties of its sub-Laplacian. This
inequality is valid on a large class of sub-Riemannian manifolds obtained from
Riemannian foliations. We give a geometric interpretation of the invariants
involved in the inequality. Using this inequality, we obtain a lower bound for
the eigenvalues of the sub-Laplacian. This inequality also lays the foundation
for proving several powerful results in Part~II.
