We prove a family of new Weitzenb\"ock formulas on a Riemannian foliationwith totally geodesic leaves. These Weitzenb\"ock formulas are naturallyparametrized by the canonical variation of the metric. As a consequence, undernatural geometric conditions, the horizontal Laplacian satisfies a generalizedcurvature dimension inequality. Among other things, this curvature dimensioninequality implies Li-Yau estimates for positive solutions of the horizontalheat equation and a sub-Riemannian Bonnet-Myers compactness theorem whoseassumptions only rely on the intrinsic geometry of the horizontal distribution.
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