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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  .doi:10.4310/ajm.2008.v12.n3.a2 fatcat:isfo3qc2hzegrjo7nhykvlqxge