Lie foliations transversely modeled on nilpotent Lie algebras
Let \(M\) be a \(n\)-dimensional closed orientable smooth manifold and let \(\mathcal F\) be a codimension \(q\) transversely orientable smooth foliation on M. Let \(g\) be some \(q\)-dimensional Lie algebra over \(\mathbb R\). The foliation \(\mathcal F\) is a Lie \(g\)-foliation if there exists a nonsingular Maurer-Cartan form \(\omega \in \Omega^1(M,g)\) such that \(T \mathcal F = Ker(\omega)\). For each Lie foliation \(\mathcal F\) there are associated two Lie algebras, the model Lie algebra \(g\) and the structure Lie algebra \(h \subset g\). There is a natural question to determine the pairs of Lie algebras \((g, h)\) which can be realized as a Lie \(g\)-foliations \(\mathcal F\) on some closed manifolds \(M\) with structure Lie algebras \(h\). Such pairs \((g,h)\) are called realizable. In this paper, two problems of realization for Lie foliations are studied: (1) Which pair of Lie algebras \((g,h)\) can be realized? and (2) Which pair \(g,m\) \((m \in \mathbb N)\) can be realized as a Lie \(g\)-foliation on a closed manifold with the structure Lie algebra \(\mathbb R^m\)? An answer to (1) in the case, when \(g\) is a nilpotent Lie algebra, is given: let \(h\) be a subalgebra of such Lie algebra \(g\), then the pair \((g, h)\) is realizable if and only if \(h\) is an ideal of \(g\) and the quotient Lie algebra \(g/h\) has a rational structure. An answer is given to (2) in the case when \(g\) is a nilpotent Lie algebra which has a rational structure: the pair \(g,m\) is realizable if and only if \(m \leq \dim c(g)\), where \(c(g)\) is the center of \(g\). Here the assumption that \(g\) has a rational structure can't be avoided.
scientific article; zbMATH DE number 6521446
scientific article