We study properties of geodesic foliations on the flat, n-dimensional torus.
Using the isomorphism of the Hodge star, we obtain some facts concerning compact totally
geodesic surfaces (which are the leaves of geodesic foliations). We compute the p-module of
a geodesic foliation. On the basis of these results, we derive a kind of reciprocity formula for
the product of modules of two orthogonal foliations. We relate this product with the number
of intersections of their leaves. We also obtain a formula for a product of modules of a finite
number of geodesic foliations.