Transverse $\mathcal F^T$-entropy and transverse Ricci flow for Riemannian foliations

TL;DR

This paper introduces a new entropy functional for Riemannian foliations that evolves monotonically under the transverse Ricci flow, linking it to gradient flow and providing conditions for transverse Einstein metrics.

Contribution

It defines a transverse $\\mathcal{F}^T$-entropy functional and establishes its monotonicity along the transverse Ricci flow, also relating it to the gradient flow and transverse Einstein metrics.

Findings

The entropy functional is monotonically decreasing along the transverse Ricci flow.

A necessary condition for codimension 4 Riemannian foliations admitting transverse Einstein metrics is provided.

The gradient flow of the entropy functional corresponds to the transverse Ricci flow via foliated diffeomorphisms.

Abstract

In this paper, we introduce an entropy functional on Riemannian foliation, inspired by the work of , which is monotonically along the transverse Ricci flow. We relate their gradient flow, via diffeomorphism preserving the foliated structure of the manifold with Riemannian foliation, to the transverse Ricci flow. Moreover, inspired by the work of Fuquan Fang and Yuhao Zhang, we give a necessary condition for codimension 4 Riemannian foliation admitting the transverse Einstein metric.