The Bochner formula for Riemannian flows

TL;DR

This paper analyzes the curvature term in the Bochner-Weitzenb{"o}ck formula for Riemannian flows, splitting it into two parts, and derives eigenvalue estimates for the basic Laplacian, characterizing the case of equality.

Contribution

It introduces a new splitting of the curvature term in the Bochner formula for Riemannian flows and provides eigenvalue bounds with geometric characterizations.

Findings

Curvature term splits into two parts: manifold curvature and O'Neill tensor.

Established lower bounds for the basic Laplacian eigenvalues.

Identified conditions under which the manifold is a local product.

Abstract

In this paper, we consider a Riemannian manifold (M, g) endowed with a Riemannian flow and we study the curvature term in the Bochner-Weitzenb{\"o}ck formula of the basic Laplacian on M. We prove that this term splits into two parts. The first part depends mainly on the curvature operator of the underlying manifold M and the second part is expressed in terms of the O'Neill tensor of the flow. After getting a lower bound for this term, depending on these two parts, we establish an eigenvalue estimate of the basic Laplacian on basic forms. We then discuss the limiting case of the estimate and prove that when equality occurs, the manifold M is a local product. This paper follows mainly the approach in [21].