Foliations with locally reductive normal bundle
We say that the foliated manifold (M,F) has a locally reductive (locally symmetric) normal bundle if its normal bundle admits a basic connection \(\nabla\) satisfying \(\nabla T=\nabla R=0 (T=\nabla R=0)\) where T and R are the torsion and curvature tensors of \(\nabla.\) Theorem 1. Let (M,F,\(\nabla)\) be a foliated manifold with a complete locally reductive normal bundle. Let p: \(\tilde M\to M\) be the universal cover of M. Then there is a simply connected reductive homogeneous space G/H and a locally trivial fiber bundle f: \(\tilde M\to G/H\) whose fibers are the leaves of \(p^{-1}(F)\). Moreover, \(p^{-1}(\nabla)=f^{- 1}({\bar \nabla})\) where \({\bar \nabla}\) is the canonical connection of the second kind on G/H. Theorem 2. Let F be a complete Riemannian locally symmetric foliation of a manifold M. If (M,F) has positive transverse sectional curvature, then M/F is compact. If in addition F has a compact leaf with finite fundamental group, then M is compact with finite fundamental group. Theorem 3. Let M and N be connected manifolds and let f: \(M\to N\) be a submersion. Let \(\nabla\) be a connection on \(Q=T(M)/\ker(f_*)\) and \({\bar \nabla}a\) linear connection on N such that \(f^{-1}({\bar \nabla})=\nabla\). If \(\nabla\) is complete, then f: \(M\to N\) is a locally trivial fiber bundle and \({\bar \nabla}\) is also complete. It is easy to see that a codimension one foliation of a compact manifold defined by a nonsingular closed one-form has a complete locally reductive normal bundle and so we obtain from Theorem 1 Reeb's structure theorem for such codimension one foliations. More generally, any Lie foliation of a compact manifold has a complete locally reductive normal bundle and so we obtain the structure theorem of Fedida.
compact leaf with finite fundamental group
scientific article; zbMATH DE number 3848246