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We develop a variational theory of geodesics for the canonical variation ofthe metric of a totally geodesic foliation. As a consequence, we obtaincomparison theorems for the horizontal and vertical Laplacians. In the case ofSasakian foliations, we show that sharp horizontal and vertical comparisontheorems for the sub-Riemannian distance may be obtained as a limit ofhorizontal and vertical comparison theorems for the Riemannian distancesapproximations.

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations