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

2014/05/16 by Yoshiaki Maeda, Maeda, Yoshiaki, Steven Rosenberg +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG

paper · pdf · doi:10.48550/arxiv.1405.4231

This supersedes the first part of arXiv:0705.1008

arxiv created 2014/05/16 · openalex publication_date 2014/05/16 · arxiv updated 2014/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Secondary Characteristic Classes on Loop Spaces

Cited by

• The Geometry of Loop Spaces V: Fundamental Groups of Geometric Transformation Groups

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• Secondary Characteristic Classes on Loop Spaces
• The Geometry of Loop Spaces II: Characteristic Classes
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