  R. Zimmer proved that, on a compact manifold, a foliation with a dense leaf,
a suitable leafwise Riemannian symmetric metric and a transverse Lie structure
has arithmetic holonomy group. In this work we improve such result for totally
geodesic foliations by showing that the manifold itself is arithmetic. This
also gives a positive answer, for some special cases, to a conjecture of E.
Ghys.
