DETERMINING LIE ALGEBRA OF TRANSVERSE FOLIATED VECTOR FIELD OF THE EXTENSION OF DENSE LEAF LIE FOLIATION ON A COMPACT CONNECTED MANIFOLD

Abstract

In $[1,2]$, we showed that any extension of a Lie $\mathcal{G}$-foliation having dense leaves on a compact connected manifold $M$ corresponds to a Lie subalgebra $\mathcal{H}$ of $\mathcal{G}$. In this paper, we determine the Lie algebra $\ell\left(M, \mathcal{F}_{\mathcal{H}}\right)$ of $\mathcal{F}_{\mathcal{H}}$-transverse foliated vector fields of an extension corresponding to the subalgebra $\mathcal{H}$. Noting by $\tilde{u}$ the $\mathcal{F}$-transverse foliated vector field associated to $u \in \mathcal{G}$, we prove that $\ell\left(M, \mathcal{F}_{\mathcal{H}}\right)=\left\{\tilde{u} / u \in \mathcal{H}^{\perp}\right.$ and $[u, h]=0$, for every $\left.h \in \mathcal{H}\right\}$

Received: May 6, 2022
Revised: August 11, 2022
Accepted: August 20, 2022