# Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

## TL;DR

This paper develops a variational framework for geodesics in totally geodesic foliations, leading to comparison theorems for sub-Laplacians, especially in Sasakian cases, connecting Riemannian and sub-Riemannian geometries.

## Contribution

It introduces a variational approach to geodesics in foliations and derives new comparison theorems for sub-Laplacians, bridging Riemannian and sub-Riemannian geometries.

## Key findings

- Comparison theorems for horizontal and vertical Laplacians.
- Limit results connecting Riemannian and sub-Riemannian distances.
- Sharp inequalities in Sasakian foliations.

## Abstract

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