Polynomial forms on affine manifolds

Abstract

An affine manifold is a differentiable manifold without boundary together with a maximal atlas of coordinate charts such that all coordinate changes extend to affine automorphisms of R These distinguished charts are called affine coordinate systems. Throughout this paper M denotes a connected affine manifold of dimension n^l. We write E for R. A tensor (field) on M is called polynomial if in all affine coordinate systems its coefficients are polynomial functions in n variables. In particular a real-valued function on M may be polynomial. It is unknown whether there exists any compact affine manifold admitting a nonconstant polynomial function. The main purpose of this paper is to prove that for certain classes of affine manifolds there is no such function. These results are then applied to demonstrate that certain polynomial forms must also vanish. For related results, see Fried, Goldman, and Hirsch [2], Fried [1], [6], and [5].