On the Completions of the Spaces of Metrics on an Open Manifold

descriptionPublicationkeyboard_double_arrow_right Article 01 May 1996 English Publisher:Springer Science and Business Media LLCJournal:Results in Mathematics, volume 29, pages 355-360 (issn: 0378-6218, eissn: 1420-9012,

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Authors: Salomonsen, Gorm;

doi: 10.1007/bf03322230 , 10.1007/bf03322530

On the Completions of the Spaces of Metrics on an Open Manifold

- Summary
- Subjects
- Metrics

Abstract

For a smooth manifold \(X\) the function \(d_0\) defined by: \[ d_0(g,g')=\sup_{x\in X}\sup_{Y\in T_x X-\{0\}} {|g_x(Y,Y)-g_x'(Y,Y)|\over g_x(Y,Y)+g_x'(Y,Y)} \] is a complete metric on the space of complete \(C^0\) metrics on \(X\). This metric is then used to show that a space of strictly positive definite smooth metrics with a natural uniform structure is complete.

Related Organizations

Aarhus University
Denmark
Max Planck Institute for Mathematics
Germany
Max Planck Society
Germany
University of Bonn
Germany

Keywords

complete metrics on the set of semi-metrics, Manifolds of metrics (especially Riemannian), complete metric space, Complete metric spaces, space of metrics

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