In [9], 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.
