An Application of Stochastic Flows to Riemannian Foliations

TL;DR

This paper develops a stochastic flow framework on the frame bundle of a Riemannian foliation, demonstrating how to perturb metrics to achieve basic-harmonic mean curvature using ergodic theory.

Contribution

It introduces a stochastic flow approach to Riemannian foliations and constructs a bundle-like metric with basic-harmonic mean curvature via ergodic methods.

Findings

Constructed a stochastic flow on the frame bundle of a Riemannian foliation.

Proved the existence of a bundle-like metric with basic-harmonic mean curvature.

Identified a unique positive function related to the generator of the transition semigroup.

Abstract

A stochastic flow is constructed on a frame bundle adapted to a Riemannian foliation on a compact manifold. The generator A of the resulting transition semigroup is shown to preserve the basic functions and forms, and there is an essentially unique strictly positive smooth function phi satisfying A^* phi = 0. This function is used to perturb the metric, and an application of the ergodic theorem shows that there exists a bundle-like metric for which the basic projection of the mean curvature is basic-harmonic.