Splitting of nonnegatively curved leaves in minimal sets of foliations
Let \(L\) be a leaf in a minimal set \({\mathcal L}\) of a smooth foliation of a compact Riemannian manifold, \(G\) an isometry group of universal covering \(\widetilde{L}\) of \(L\). The authors prove, that 1) if \(G\) is cocompact (\(\widetilde{L}= G*K\) for some compact subset \(K\subseteq L\)) then there exist: (A) a compact Hausdorff space \(X\), (B) a locally free minimal action of \(G\) on \(X\), (C) a continuous map \(\eta: X\to {\mathcal L}\) which carries \(G\)-orbits to leaves of \({\mathcal L}\), such that the image of \(\eta\) intersects every leaf of \({\mathcal L}\), and for every \(x\in X\), the leaf through \(\eta(x)\) is isometric to \(\widetilde{L}/ G_ x\), where \(G_ x\) denotes the stabilizer in \(G\) of \(x\); 2) if \(L\) has nonnegtive Ricci curvature, then \(\widetilde{L}\) is isometric to the Riemannian product of a compact manifold by a flat Euclidean space, moreover, any two leaves of \({\mathcal L}\) a) have isometric universal cover, b) which split are isometric, and some other statements. The proofs are based on the theory of measure-preserving and topological actions, Cheeger and Gromoll splitting theorem and the results of the first author and \textit{A. Freire} [J. Differ. Geom. 34, No. 3, 681-700 (1991; Zbl 0751.53010)].
scientific article; zbMATH DE number 495437
scientific article