Structures géométriques invariantes et feuilletages de Lie. (Invariant geometric structures and Lie foliations) (Q748917)

Let G be a connected Lie group and K a subgroup of G such that the homogeneous space \(W=G/K\) is compact. Suppose G acts on a vector bundle E over a manifold M. This action induces a natural action of G on the space \(F=C^{\infty}(E)\). The authors show that if \(W=G/K\) is amenable, then there exists an affine and continuous application m: \(F_ K\to F_ G\) such that: (i) \(m(\alpha)=\alpha\) for each \(\alpha \in F_ G\), (ii) if A is an open convex subset of F, such that, for each \(\alpha \in F_ K\cap A\), the orbit of \(\alpha\) is contained in A, then \(m(F_ K\cap A*)\subset F_ G\cap A.\) Here \(F_ K\) and \(F_ G\) are the subspaces of F containing K-invariant and G-invariant sections, respectively. The above fact has numerous consequences. For example: if M (with an action of G) is complex and admits a K-invariant Kähler or symplectic structure, then it admits the respective G-invariant structure. If \(M=G\) and acts on itself by left translations, then the existence of some geometric structures on G implies some algebraic properties of G. The second part of the paper is devoted to the relations between the cohomology \(H^*({\mathcal G})\) of the Lie algebra \({\mathcal G}\) of G and the cohomology \(H^*_ K(G)\) of K-invariant forms on G. Some description of the Lie foliation on a compact manifold is given.