This paper has its roots in the well-known work of Reeb and its recent generalization by Reinhart [9], [10], as well as in much older, incomplete work of E. Cartan dealing with the differential-geometric structure of a foliation [3]. The general problem is to see how the structure affects the global properties of the foliation. The foliations with "bundle-like" metrics, as studied by Reinhart, seem to offer the most interesting field of application of this point of view. As our main result in this direction, we can state: