The Completion of the Manifold of Riemannian Metrics

2009/04/01 by Brian Clarke, Clarke, Brian · 1 citation
Mathematics · Physics and Astronomy · #58D17 (Primary) 58B20 (Secondary) #Advanced Differential Geometry Research #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #math.DG #msc:58B20 #msc:58D17

paper · pdf · doi:10.48550/arxiv.0904.0177

43 pages

arxiv created 2009/04/01 · openalex publication_date 2009/04/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a description of the completion of the manifold of all smooth Riemannian metrics on a fixed smooth, closed, finite-dimensional, orientable manifold with respect to a natural metric called the L² metric. The primary motivation for studying this problem comes from Teichmueller theory, where similar considerations lead to a completion of the well-known Weil-Petersson metric. We give an application of the main theorem to the completions of Teichmueller space with respect to a class of metrics that generalize the Weil-Petersson metric.

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