A global lifting for finite-dimensional Lie algebras of complete vector fields

Motivated by the so-called Lifting and Approximation Theorem by Rothschild and Stein, we consider a set of vector fields X = {X1, . . ., Xn} on a manifold M, and we study the problem of obtaining a global lifting of the Xi’s to a system of generators of the Lie algebra Lie(G) of a Lie group G. By assuming that the Lie algebra g generated by X is finite-dimensional and all of the Xi’s are complete vector fields, but without the assumption that they satisfy Hörmander’s rank condition, we reduce the lifting problem to a result of Palais on integrability. This proves that any X in g is E-related to a left invariant vector field X ∈ Lie(G), where E : G → M is a smooth map resulting from a right action θ of G on M. Both the lifting map E and the lifting vector fields X are globally defined, and our result generalizes the global Lifting obtained by Folland in the special case of dilation-invariant vector fields. According to Palais’ Integrability, the map E is obtained via the flow of suitable vector fields on M and the image set of E, namely, θ(x)(G), is the Sussmann orbit of g through x ∈ M. Examples are provided, showing that a germ of E, obtained through the integration of g and depending only on the Baker–Campbell–Hausdorff formula, is often sufficient to get the global lifting without the need of abstract results.