Every surface is a leaf
Let L denote an open, connected 2-manifold, M a closed, connected 3- manifold. The main theorem of the paper asserts that if L is orientable and M is arbitrary, or if L and M are both nonorientable, then there exists a transversely orientable, \(C^{\infty}\) foliation of M having a leaf diffeomorphic to L. Directly from the construction in the proof of this theorem the authors obtain the following results: (1) Every open surface can be realized as a leaf with exponential growth in suitable closed, \(C^{\infty}\)-foliated 3-manifolds. (2) There exists a transversely orientable, \(C^{\infty}\) foliation of M and a Riemannian metric on M such that a certain leaf \(L_*\) lies at infinite level, is diffeomorphic to L, and has constant curvature -1. The authors also assert that the assumption of compactness of M in the main theorem can be omitted.
scientific article; zbMATH DE number 4007265