On the Moser trick for Lie subalgebras and foliations

Technical Summary: On the Moser Trick for Lie Subalgebras and Foliations

Problem Statement
The paper addresses the problem of characterizing the smooth triviality of deformations within the context of Lie algebroids. Specifically, it focuses on two limiting cases of Lie algebroids: Lie subalgebras and foliations. While the deformation theory for Lie algebroids has been established (notably by Crainic and Moerdijk), the author seeks to provide direct, self-contained proofs for these specific corner cases. The central question is to establish necessary and sufficient conditions under which a smooth deformation of a Lie subalgebra or a foliation is "trivial" (i.e., equivalent to the constant deformation via a smooth family of automorphisms or diffeomorphisms). A particular motivation is to extend these results to general Lie subalgebroids where the quasi-isomorphism between different deformation complexes (which holds for foliations) generally fails.

Methodology
The author employs a deformation-theoretic approach rooted in the Chevalley-Eilenberg complex and the "Moser trick" (a method involving time-dependent vector fields and flows to construct equivalences).

1. Lie Subalgebras and Ideals (Section 2):

   • The author associates a family of cocycles to a smooth deformation of a Lie algebra morphism, a Lie subalgebra, or a Lie ideal.
   • By differentiating the defining equations of these structures with respect to the deformation parameter $t$, the author derives conditions involving the time-derivative of the deformation maps.
   • The core technique involves constructing time-dependent vector fields on the relevant Lie groups or vector spaces. The existence of a solution to a specific differential equation (involving the deformation cocycle and a time-dependent vector field) is shown to be equivalent to the triviality of the deformation.
   • The proofs rely on the local existence of flows for time-dependent vector fields and the uniqueness of integral curves.

2. Foliations (Section 3):

   • The paper treats a foliation as an involutive distribution. A smooth deformation is defined as a foliation on $M \times I$ tangent to the slices.
   • The author utilizes the deformation complex of foliations, identified with $(\Omega^\bullet(\mathcal{F}; \nu(\mathcal{F})), d^\nabla)$, where $\nu(\mathcal{F})$ is the normal bundle and $\nabla$ is the Bott connection.
   • A "deformation cocycle" $\sigma_t$ is defined as the time-derivative of the projection onto the foliation.
   • The Moser trick is implemented by constructing a time-dependent vector field on the product manifold $M \times I$. The involutivity of a specific subbundle $D$ (constructed from the total foliation and the vector field) is shown to be equivalent to the vanishing of the deformation cocycle in cohomology.

Key Contributions and Results

• Lie Subalgebras (Theorem 2.2): The paper establishes a cohomological criterion for the triviality of a smooth deformation of a Lie subalgebra $(h, \iota) \subset g$. It proves that a deformation is $Aut(g)$-trivial if and only if the associated cochain $\pi_{can} \circ \dot{\iota}_t$ (the projection of the time-derivative of the inclusion map) vanishes in the cohomology of the deformation complex $C^\bullet_{\iota_t}(h_t; g/\iota_t(h_t))$.
• Lie Ideals (Theorem 2.3): An analogous criterion is derived for the deformation of a Lie ideal within a short exact sequence of Lie algebras. The triviality of the deformation is linked to the vanishing of a specific cocycle in the complex $C^\bullet_{\pi_t}(g; Hom(h_t, k_t))$.
• Moser Trick for Foliations (Theorem 3.7): The author provides a direct proof of the Moser trick for foliations. The main result states that a smooth deformation $(F_t)_{t \in I}$ of a foliation $F$ on a compact manifold $M$ is trivial if and only if the deformation cocycles $\sigma_t$ vanish smoothly in cohomology (i.e., $[\sigma_t] = 0$ in $H^1(F_t; \nu(F_t))$).
• Refinement of Existing Results (Lemma 3.8): The paper demonstrates that a trivial deformation always yields an exact deformation cocycle, recovering a result from Heitsch [10] but under weaker assumptions (removing the need for a complete vector field on the ambient manifold).

Significance and Claims
The paper claims to provide a "direct proof" of the Moser trick for foliations that relies solely on the deformation complex of the foliation viewed as an involutive distribution, rather than invoking the broader machinery of Lie algebroid deformation theory.

The author notes that while the deformation theory for Lie algebroids (Crainic-Moerdijk) implies the result for foliations via a quasi-isomorphism, this quasi-isomorphism does not generally hold for arbitrary Lie subalgebroids. Therefore, the techniques developed here—specifically the infinitesimal analogues of Moser tricks for Lie subalgebras and ideals—are intended to serve as a foundation for extending the triviality results to general Lie subalgebroids, a task the author indicates is the subject of subsequent work [1].

The work is modest in its scope, focusing on establishing the necessary and sufficient cohomological conditions for triviality in these specific geometric settings without proposing new experimental applications or broad physical theories. It serves as a technical bridge between the deformation theory of Lie algebras and the more complex theory of Lie algebroids.