  In this paper, we introduced 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 diffeomorphisms preserving the
foliated structure of the manifold with Riemannian foliation, to the transverse
Ricci flow. Moreover, inspired by the work of Fang and Zhang, we give a
necessary condition for codimension 4 Riemannian foliation admitting the
transverse Einstein metric.
