Dynamics and the Godbillon–Vey class of $C^1$ foliations

Abstract

Let F be a codimension-one, C 2 -foliation on a manifold M without boundary. In this work we show that if the Godbillon-Vey class GV (F ) ∈ H 3 (M ) is non-zero, then F has a hyperbolic resilient leaf. Our approach is based on methods of C 1 -dynamical systems, and does not use the classification theory of C 2 -foliations. We first prove that for a codimension-one C 1foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points E(F ) has positive Lebesgue measure. We then prove that if E(F ) has positive measure for a C 1 -foliation, then F must have a hyperbolic resilient leaf, and hence its geometric entropy must be positive. The proof of this uses a pseudogroup version of the Pliss Lemma. The first statement then follows, as a C 2 -foliation with non-zero Godbillon-Vey class has non-trivial Godbillon measure. These results apply for both the case when M is compact, and when M is an open manifold.

A Godbillon-Vey type invariant for a 3-dimensional manifold with a plane field

“…If F is a Riemannian foliation and g is bundle-like, then the normal vector field T is geodesic (i.e., ∇ T T = 0) and η = 0, see Lemma 6 again. Thus, dη = 0, and (T F, T ) is critical by (9). 6 Variable Riemannian metric Functional (28) leads to two functionals on the space of metrics Riem(M ) on a manifold M 3 equipped with either a plane field D (then T varies) or a unit vector field T (then D varies).…”

“…The Godbillon-Vey class plays a crucial role in the study of topology and dynamics of foliations and, still, is of some interest among "foliators", see e.g. [2,5,7,9,16] and [8,Problem 10]. Now, let g be a Riemannian metric on M (dim M = 3), ∇ its Levi-Civita connection, T the positive oriented unit vector field on M normal to F and h the scalar second fundamental form.…”

Losik classes for codimension-one foliations

“…From (11) it follows that, for a given open subset U ⊂ R and coordinates (12), the coordinates (16) x 0 , x 2 , x 3 , . .…”

“…where we used the coordinates (10) on S(U ) for an arbitrary open subset U ⊂ R. These forms are well-defined also on S ′′ (X) and may be written with respect to the coordinates (12) in the following way:…”

“…Despite the simple description, its geometrical meaning remains a mystery. There are many works (e.g., [2,5,6,20,21,28]) studying the relation of (non-)triviality of the Godbillon-Vey class to the geometry and dynamics of the foliation, see the surveys [11,12] and the book [3] for the complete list of references. It is natural to ask for stronger invariants that could provide more information about the geometry of the foliation, e.g., in [5,11], it was suggested to study the Vey class [dη] introduced by Duminy and taking values in certain cohomology associated to the foliation.…”