  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.
