The Lie affine foliations on 4-manifolds
A Lie affine foliation \({\mathcal H}\) on a closed manifold \(M^ n\) is a codimension 2 foliation locally modelled transversely on the group \(\text{Aff}_ +(\mathbb{R})\) of all orientation preserving affine transformations of the real line. Equivalently, the foliation \({\mathcal H}\) is defined by the equation \(\omega= \eta=0\) for a pair of everywhere independent 1-forms \(\omega\) and \(\eta\) which satisfy the equations \(d\omega= \omega\wedge\eta\) and \(d\eta=0\). If the closed 1-form \(\eta\) is rational, then the manifold \(M^ n\) is the total space of a fiber bundle over the circle and the 1-form \(\omega\) restricted to each fiber is closed. In the converse way, given an \((n-1)\)-manifold \(N^{n-1}\), a non-singular 1-form \(\omega_ 0\) on \(N\) and a diffeomorphism \(F:N\to N\) such that \(F^*\omega_ 0= \exp(\lambda)\omega_ 0\), one can define a Lie affine foliation on the total space of the \(N\)-bundle over the circle with the monodromy map \(F\), whose restriction to each fiber is the foliation defined by \(\omega_ 0\). If the dimension \(n\) is 3, it is known that all the Lie affine foliations are obtained in this way. In the present paper, the authors treat the case \(n=4\) and prove that all the Lie affine foliations are equivalent to some foliation obtained in the above way where \(F\) and \(\omega_ 0\) are defined explicitly in an algebraic manner. More precisely, \(N\) is a quotient of simply connected 3-dimensional nilpotent Lie group by a uniform lattice and the closed form \(\omega_ 0\) is an invariant form under the natural action of the group and \(F\) is induced from an automorphism of the Lie group. In the proof, the authors show first the manifold \(M\) has a bundle structure over the circle with some 3-manifold \(N\) as fiber. Then they investigate closely this 3-manifold and the foliation induced on it. Some results on 3-manifolds such as Jaco-Shalen-Johanson decomposition, Mostow-Sullivan rigidity theorem are essentially used.
orientation preserving affine transformations of the real line
total space of a fiber bundle over the circle
scientific article; zbMATH DE number 126368
scientific article