Tangentially affine foliations and leafwise affine functions on the torus
The authors study tangentially affine foliations \({\mathcal F}\) of the torus \(T^ 2\) by curves. That is, there is a distinguished atlas in which the coordinate transformations are affine in the \({\mathcal F}\)-direction. They prove that such foliations exist and that any leafwise affine function \(F:T^ 2\to\mathbb{R}\) for \({\mathcal F}\) is uniquely determined by its values on the union of the compact leaves. Furthermore, if \(L\) is a compact leaf with contracting holonomy on one side and if \(U\) is a small neighborhood of \(L\) on that side, they show that the vector space \({\mathcal A}\) of all leafwise affine functions on \(U\) has dimension 1, 2, or \(\infty\). Finally, in a closing remark they show that the Reeb foliation of \(S^ 3\) admits no tangentially affine structure.
scientific article; zbMATH DE number 409640