The Geometry of Loop Spaces I: $H^s$-Riemannian Metrics

Abstract

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter s. We compute the connection forms of these metrics and the higher symbols of their curvature forms, which take values in pseudodifferential operators. These calculations are used in a followup paper "The Geometry of Loop Spaces II: Characteristic Classes" to construct Chern-Simons classes on the tangent bundle TLM which detect nontrivial elements in the diffeomorphism group of certain Sasakian 5-manifolds associated to Kaehler surfaces.