Equivariant Morse theory for Lie algebra actions on Riemannian foliations

We consider a transverse isometric action of a finite-dimensional Lie algebra $\mathfrak g$ on a Riemannian foliation. In this setting, we study equivariant Morse--Bott theory on the leaf space of the foliation. Among other results, we establish a foliated Morse--Bott lemma for $\mathfrak g$-invaria