A characterization of harmonic foliations by the volume preserving property of the normal geodesic flow

Hobum Kim

doi:10.1155/s0161171202007822

Abstract

We prove that a Riemannian foliation with the flat normal connection on a Riemannian manifold is harmonic if and only if the geodesic flow on the normal bundle preserves the Riemannian volume form of the canonical metric defined by the adapted connection.

Highlights

Let (M, gM ) be a Riemannian manifold
A Riemannian foliation is harmonic if and only if either one of the following conditions holds: (1) it is an extremal of the energy functional for special variations; (2) it is an extremal of the energy of the foliation under certain variations of the Riemannian metric of the manifold
We give a dynamical characterization of the harmonicity of a Riemannian foliation which has the flat normal connection in the sense of Oshikiri [4]

Summary

Introduction

Let (M, gM ) be a Riemannian manifold. A foliation Ᏺ on M is Riemannian and gM bundle-like if all the leaves are locally equi-distant to each other. We give a dynamical characterization of the harmonicity of a Riemannian foliation which has the flat normal connection in the sense of Oshikiri [4].

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Results

Conclusion