Polynomial forms on affine manifolds

William Goldman and Morris William Hirsch

Vol. 101 (1982), No. 1, 115–121

DOI: 10.2140/pjm.1982.101.115

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 Rⁿ. These distinguished charts are called affine coordinate systems.

Throughout this paper M denotes a connected affine manifold of dimension n ≧ 1. We write E for Rⁿ.

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].

Mathematical Subject Classification 2000

Primary: 53C10

Secondary: 53C15, 57R15, 58A10

Milestones

Received: 10 June 1981

Published: 1 July 1982