L P -Chen forms on loop spaces

Abstract

INTRODUCTION Let M be a Riemannian manifold. Our aim is to study differential forms on the following infinite dimensional manifolds: (1) the path space PM consisting of paths w : [0, 1] → M , (2) the loop space LM consisting of paths w such that w (0) = w (1), (3) the based loop space L x M consisting of loops w such that w (0) = w (1) = x where x is a chosen base point in M . One consequence of the fact that these manifolds are infinite dimensional is that there are infinite sequences α n of forms with each α n homogeneous of degree n . These infinite sequences are very important in the geometrical applications of loop spaces; for example they are essential in the theory of equivariant cohomology in infinite dimensions as is made quite clear in [25]. In [11] Chen describes the theory of “iterated integrals”; this is a method of constructing differential forms on these infinite dimensional manifolds. We will refer to forms constructed by this means as Chen forms. The purpose of this paper is to study some of the analytical properties of Chen forms; in particular to make estimates for suitable L P -norms and to consider various decay conditions which one might put on the terms in an infinite sequence of the kind mentioned in the previous paragraph.