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Foliation de Rham cohomology of generic Reeb foliations

Yong-Geun Oh

TL;DR

The paper demonstrates a generic triviality of the foliation de Rham cohomology for Reeb foliations: for a residual set of contact forms $\lambda$ on a compact connected manifold $M$, both $H^0(\mathcal{F}_\lambda)$ and $H^1(\mathcal{F}_\lambda)$ are one-dimensional. This is reformulated equivalently as the unique solvability of the functional equation $R_\lambda[f]=u$ (mod constants) for mean-zero $u$, and as a reduction of the Lie algebra of strict contactomorphisms to the span of the Reeb vector field $R_\lambda$. The proof blends contact Hamiltonian dynamics with control theory and global transversality: it introduces a moduli space of Reeb trajectories, analyzes the discriminant $g_{(H;\lambda)}$, and applies Sard–Smale type arguments to achieve generic nondegeneracy. The results also provide a key ingredient toward the generic scarcity of strict contactomorphisms, connecting dynamical, cohomological, and geometric aspects of contact manifolds to establish a robust, operator-theoretic and cohomological framework for understanding generic Reeb foliations.

Abstract

In this paper, we prove that there exists a residual subset of contact forms $λ$ (if any) on a compact connected orientable manifold $M$ for which the foliation de Rham cohomology of the associated Reeb foliation $F_λ$ is trivial in that both $H^0(F_λ,{\mathbb R})$ and $H^1(F_λ,{\mathbb R})$ are isomorphic to $\mathbb R$. We also prove the same triviality for a generic choice of contact forms with fixed contact structure $ξ$. For any choice of $λ$ from the aforementioned residual subset, this cohomological result can be restated as any of the following two equivalent statements: (1) The functional equation $R_λ[f] = u$ is uniquely solvable (modulo the addition by constant) for any $u$ satisfying $\int_M u\, dμ_λ=0$, or (2) The Lie algebra of the group of strict contactomorphisms is isomorphic to the span of Reeb vector fields, and so isomorphic to the 1 dimensional abelian Lie algebra $\mathbb R$. This result is also a key ingredient for the proof of the generic scarcity result of strict contactomorphisms by Savelyev and the author.

Foliation de Rham cohomology of generic Reeb foliations

TL;DR

The paper demonstrates a generic triviality of the foliation de Rham cohomology for Reeb foliations: for a residual set of contact forms on a compact connected manifold , both and are one-dimensional. This is reformulated equivalently as the unique solvability of the functional equation (mod constants) for mean-zero , and as a reduction of the Lie algebra of strict contactomorphisms to the span of the Reeb vector field . The proof blends contact Hamiltonian dynamics with control theory and global transversality: it introduces a moduli space of Reeb trajectories, analyzes the discriminant , and applies Sard–Smale type arguments to achieve generic nondegeneracy. The results also provide a key ingredient toward the generic scarcity of strict contactomorphisms, connecting dynamical, cohomological, and geometric aspects of contact manifolds to establish a robust, operator-theoretic and cohomological framework for understanding generic Reeb foliations.

Abstract

In this paper, we prove that there exists a residual subset of contact forms (if any) on a compact connected orientable manifold for which the foliation de Rham cohomology of the associated Reeb foliation is trivial in that both and are isomorphic to . We also prove the same triviality for a generic choice of contact forms with fixed contact structure . For any choice of from the aforementioned residual subset, this cohomological result can be restated as any of the following two equivalent statements: (1) The functional equation is uniquely solvable (modulo the addition by constant) for any satisfying , or (2) The Lie algebra of the group of strict contactomorphisms is isomorphic to the span of Reeb vector fields, and so isomorphic to the 1 dimensional abelian Lie algebra . This result is also a key ingredient for the proof of the generic scarcity result of strict contactomorphisms by Savelyev and the author.

Paper Structure

This paper contains 37 sections, 61 theorems, 391 equations.

Table of Contents

  1. Introduction
  2. Strict contactomorphisms
  3. Statement of main theorems
  4. Generic triviality of the foliation de Rham cohomology
  5. Overall strategy of the proof: some control theory
  6. Basic contact Hamiltonian geometry and dynamics
  7. The conformal exponent of a contactomorphism
  8. Hamiltonian calculus of contact vector fields
  9. Contact triad, triad metric and connection
  10. Geometry of one-jet bundle of contact manifold
  11. Cohomological formulation of the functional equation $R_\lambda[f] = u$
  12. Necessary condition for the solvability
  13. Foliation de Rham cohomology of Reeb foliations
  14. Unique solvability of the equation $R_\lambda[f] = u$
  15. Big and small kinetic theory phase spaces
  16. ...and 22 more sections

Key Result

Lemma 1.5

Suppose $H = H (x)$ satisfies $R_\lambda[H] = 0$. Then the Hamiltonian flow $\psi_H^t$ is a strict contact flow and vice versa. Equivalently speaking, where $\mathfrak{cont}^{\text{\rm st}}(M,\lambda)$ is the Lie algebra of the group of $\lambda$-strict contactomorphisms.

Theorems & Definitions (109)

  • Definition 1.1: Contact pair
  • Definition 1.2: Discriminant of a contact pair
  • Definition 1.3: Strict contact pair
  • Remark 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Proposition 1.8: Proposition \ref{['prop:L2-adjoint']}
  • Corollary 1.9
  • Theorem 1.10
  • ...and 99 more