On the geometry of free loop spaces

Abstract

We verify the following three basic results on the free loop spaceLM. (1) We show that the set of all points, where the fundamental form onLMis nondegenerate, is an open subset. (2) The connections of a Fréchet bundle overLMcan be extended toS1-central extensions and, in particular, there exist natural connections on the string structures. (3) The notion of Christoffel symbols and the curvature are introduced onLMand they are described in terms of Christoffel symbols ofM.