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We give a generalized curvature-dimension inequality connecting the geometryof sub-Riemannian manifolds with the properties of its sub-Laplacian. Thisinequality is valid on a large class of sub-Riemannian manifolds obtained fromRiemannian foliations. We give a geometric interpretation of the invariantsinvolved in the inequality. Using this inequality, we obtain a lower bound forthe eigenvalues of the sub-Laplacian. This inequality also lays the foundationfor proving several powerful results in Part~II.

Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations, Part I