Rigidity of Lie affine foliations

In previous work by El Kacimi Alaoui-Guasp-Nicolau, a cohomological criterion is given for a Lie $\mathfrak{g}$-foliation on a compact manifold to be rigid among nearby Lie foliations. Our aim is to look for examples of this rigidity statement in case the Lie foliation is modeled on the two-dimensional non-abelian Lie algebra $\mathfrak{g}=\mathfrak{aff}(1)$. We study the relevant cohomology group in detail, showing that it can be expressed in terms of Morse-Novikov cohomology. We find the precise conditions under which it vanishes, which yields many examples of rigid Lie affine foliations. In particular, we show that any Lie affine foliation on a compact, connected, orientable manifold of dimension $3$ or $4$ is rigid when deformed as a Lie foliation. Our results rely on a computation of the Morse-Novikov cohomology groups associated with a nowhere-vanishing closed one-form with discrete period group, which may be of independent interest.