Foliations with leaves of nonpositive curvature
The author answers a question of \textit{M. Gromov} [Geom. Funct. Anal. 1, 253-320 (1991; Zbl 0768.53012)] in the codimension one case: if \(F\) is a codimension one foliation of a compact manifold \(M\) with leaves of negative curvature, then \(\pi_1(M)\) has exponential growth. He proves a result analogous to \textit{R. J. Zimmer} [Am. J. Math. 105, 1011-1022 (1983; Zbl 0527.57015)]: if \(F\) is a codimension one foliation on a compact manifold \(M\) with leaves of nonpositive curvature, and if \(\pi_1(M)\) has subexponential growth, then almost every leaf is flat. Finally, he gives a foliated version of the Hopf theorem on surfaces without conjugate points: if \(F\) is an orientable foliation on a compact manifold \(M\) with 2-dimensional leaves having no conjugate points, if the Euler class of the tangent bundle of the foliation is zero in real homology, and if all leaves have subexponential growth, then all leaves are flat.
scientific article; zbMATH DE number 1011392
scientific article