Totally geodesic Riemannian foliations with locally symmetric leaves (Q819847)

The paper under review deals with a Riemannian manifold \(M\) equipped with a foliation \(F\). The Riemannian metric is locally fibration-like. The leaves of \(F\) are totally geodesic and are covered isometrically by the same symmetric space \(X_G=G\slash K\) that is the quotient of a noncompact semisimple Lie group \(G\) by its maximal compact subgroup \(K\). The foliation \(F\) is supposed to have a dense leaf. The main Theorem 1 of the paper states that under the above assumptions the foliation \(F\) has arithmetic nature. Namely, the foliated manifold \(M\) admits an isometric covering by a product \(\widetilde M=X_G\times Y\) of homogeneous spaces, \(Y=H\slash L\) is the quotient of a semisimple Lie group \(H\) by some compact subgroup \(L\); there exists an arithmetic lattice \(\Gamma\in G\times H\) such that the left quotient \(\Gamma\setminus\widetilde M\) of the previous product covering by the lattice \(\Gamma\) is a finite covering of the foliated space \(M\); the horizontal foliation of \(\widetilde M\) by copies of \(X_G\) is projected onto \(F\). Theorem 2 says that the above arithmetic quotient \(M_1=\Gamma\setminus\widetilde M\) admits a fibration \(\pi:M_1\to M_2\) with compact fibers over an irreducible symmetric space \(M_2\) of noncompact type. The latter projection transforms the foliation on \(M_1\) (induced by the previous horizontal foliation on \(\widetilde M\)) to a foliation on \(M_2\). The proof of Theorem 1 is based on a previous result of the author [\textit{R. Quiroga-Barranco}, C. R., Math., Acad. Sci. Paris 341, No. 6, 361--364 (2005; Zbl 1076.53087)]. (cited as Theorem 3.2 in the paper under review) that provides a result similar to Theorem 1 for foliations obtained as orbits of a locally free \(G\)-action. Such action preserves a finite volume pseudo-Riemannian metric that induces a transverse Riemannian structure so that the geodesics orthogonal to the \(G\)- orbits are complete. The \(G\)-action is faithful and has a dense orbit. For the proof of Theorem 1 the author considers the auxiliary space \(M^*\) of isometric coverings of the leaves of the foliation \(F\) by the symmetric space \(X_G\). The manifold \(M^*\) is fibered over \(M\) and admits a locally free action by \(G\); its orbits are projected to the leaves of \(F\). The author shows that the new foliated manifold \(M^*\) satisfies the conditions of the above-mentioned Theorem 3.2 and applies this Theorem.