Stability for a Class of Foliations Covered by a Product

摘要

In this paper, we study transversely orientable, codimension-1 C~1 foliations of Riemannian 3-manifolds. In particular, we examine foliations of a manifold M that are covered by the canonical foliation of R~3 by parallel hyperplanes, which we refer to as "covered by a product". These foliations are particularly nice in the sense that they are completely determined by the induced action of π_1 (M) on the real line, which is the leaf space of the universal cover (see [20]). This family of foliations includes fibrations as well as weak stable (or unstable) foliations associated with many Anosov flows, including geodesic flows or suspensions of Anosov diffeomorphisms. Several recent works have focused on whether the associated foliations of other Anosov flows are covered by a product (e.g., [1; 3; 5]).