Leafwise cohomology and rigidity of certain Lie group actions

Abstract

A locally free Lie group action on a closed manifold is called parameter rigid, if given another action with the same orbit foliation, there is a C ∞ orbit-preserving diffeomorphism which conjugates one action to the other, up to an automorphism of the Lie group. We show some relationships between parameter rigidity and the first leafwise cohomology of the orbit foliation, and give a new example of a parameter rigid solvable group action. We also compute the first leafwise cohomology of the stable foliation of geodesic flows of closed orientable surfaces of constant negative curvature. IntroductionThe leafwise cohomology H * (F ) of a C ∞ foliation F on a manifold M is the cohomology of the de Rham complex formed by the leafwise C ∞ forms (i.e. C ∞ sections of the exterior product of the cotangent bundle of the foliation).This complex is not elliptic, and the leafwise cohomology can be infinite-dimensional in general. For example, if the manifold M is the product L × T of two manifolds L and T and if the foliation F is given as the product foliation {L × {t} | t ∈ T }, then the leafwise cohomology H i (F ) turns out to coincide with the space C ∞ (T , H i (L; R)) and is infinite-dimensional unless H i (L; R) vanishes.Alvarez López and Hector [AH] have given sufficient geometric conditions for foliations (especially for foliations with dense leaves on compact manifolds) to have infinite-dimensional leafwise cohomology. In fact they are concerned with the infinite dimensionality of the so-called reduced leafwise cohomology.On the other hand, a linear one-dimensional foliation on T n with badly-approximable slope [Sc] was shown to have one-dimensional first leafwise cohomology [He].

Parameter rigid actions of simply connected nilpotent Lie groups

“…Proposition 3 is obtained by examining the proof of Corollary 2 in [4]. In this paper, we will identify a cocycle with its corresponding differential form.…”

Parameter rigid actions of the Heisenberg groups

“…For R n -actions, CR1 and PR are equivalent properties [8]. We show in §1 (Proposition 1.3) that a CR1 G-action A leaves invariant a smooth volume form , i.e.…”

Nonhomogeneous locally free actions of the affine group

“…Here, we present a shorter proof using a rigidity result due to Matsumoto and Mitsumatsu [17]. Fix a C ∞ locally free action ρ on M .…”

“…Since the orbit foliation of ρ admits no closed leaves, a classification result by Ghys and Sergiescu [4] implies that F ρ is C ∞ diffeomorphic to the orbit foliation F A of a homogeneous GA-action ρ A on M A . In [17], Matsumoto and Mitsumatsu proved that any locally free GA-action on M A whose orbit foliation is F A is C ∞ conjugate to ρ A . Therefore, ρ is C ∞ conjugate to the homogeneous action ρ A .…”