One-dimensional Riemannian foliations on the Heisenberg group.

Abstract

The main goal of our dissertation is to show that one-dimensional Riemannian foliations on the Heisenberg group H2 n+1 are homogeneous. Using our result and the homogeneity of codimension one Riemannian foliations on H 2n+1 proved by G. Walschap, we conclude that every Riemannian foliation on the three-dimensional Heisenberg group is homogeneous. As noted in the literature, the result mentioned above remains valid on Gamma\H3, where Gamma is a lattice in H3. We give another proof for this theorem and we improve it by showing that there are no codimension one foliations on Gamma\H2n +1. We also show that the only one-dimensional Riemannian foliation on the space above occurs as the projection of the foliation on H 2n+1 with leaves tangent to the center.