Linear foliations ofT n

Abstract

We present a definition of diophantine matrix and use this concept to distinguish two classes of minimal linear foliations ofTn, the diophantine and the Liouville one. Let e p , 1≤p≤n−1, denote a minimal (all leaves are dense) linearp-dimensional foliation ofTn, andHom(Tn, e p ), 1≤m≤p, the cohomology group of type (0,m) of the foliated manifold (Tn, e p ). Our main result is the computation of these groups.Hom(Tn, e p ) is isomorphic to\(\mathbb{R}^{(\begin{array}{*{20}c} n \\ m \\ \end{array} )} \) if e p is diophantine and is an infinite dimensional non-Hausdorff vector space if e p is Liouville. Some of these groups were computed before, see [4], [6] and [9].