Arithmeticity of Totally Geodesic Lie Foliations with Locally Symmetric Leaves

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.

Keywords

Semisimple Lie groups, arithmeticity, foliations, transverse structures, tangential structures, pseudoRiemannian geometry

2010 Mathematics Subject Classification

Primary 53C12, 53C24. Secondary 22E46, 53C10.

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Published 1 January 2008