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

TL;DR

This paper investigates the geometric structure of loop spaces endowed with Sobolev-type Riemannian metrics, computing connection forms and curvature symbols valued in pseudodifferential operators, laying groundwork for characteristic class construction.

Contribution

It introduces explicit calculations of connection and curvature forms for Sobolev Riemannian metrics on loop spaces, advancing understanding of their geometric and topological properties.

Findings

Computed connection forms of Sobolev Riemannian metrics on loop spaces

Derived higher symbols of curvature forms as pseudodifferential operators

Provided foundational results for constructing characteristic classes in subsequent work

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.