Arithmeticity of Totally Geodesic Lie Foliations with Locally Symmetric Leaves

Abstract

Abstract. Zimmer [9] 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 [5].

Isometric actions of simple groups and transverse structures: The integrable normal case

“…We now obtain the next result which describes actions preserving a metric and a transverse parallelism. Its proof is based on some of the arguments found in [11]. Proposition 6.2.…”

“…In Section 6 we establish the characterization of G-spaces of the form (G × H)/Γ provided by Theorem 1.3; for this, one of the main ingredients is given by Theorem 1.1. We observe that the arguments used in Section 6 are based on those found in [11].…”