Among different topological types of foliations, compact foliations on compact manifolds, i.e., foliations with all leaves compact, from our point of view, are very interesting. In 1952, Reeb (Actual scient. ind. 1183:93–154, 1952) described a smooth flow on non-compact manifold which has periodic orbits such that the length of orbits is locally unbounded. Edwards et al. (Topology, 16:13–32, 1977) have given a full answer to the following question (named the Periodic Orbit Conjecture). If M is a compact manifold foliated by compact submanifolds, is there an upper bound on the volume of the leaves? In general, the answer is negative, as the following sections show. First, we present two examples of foliations by circles of compact manifolds for which the length of the leaves is unbounded. With volume function, we study the topology of the leaf space of compact foliations. The last section is devoted to the Periodic Orbit Conjecture, for which we present the answer in codimension two.
KeywordsVolume Function Compact Manifold Riemannian Structure Full Answer Compact Foliation
- 14.Reeb, G.: Sur certaines properiétés topologiques des variétés feuilletées. Actual scient. ind. 1183, 93–154 (1952)Google Scholar