Open AccessOn Certain Geometric Operators Between Sobolev Spaces of Sections of Tensor Bundles on Compact Manifolds Equipped with Rough Metrics
Author(s) -
A. Behzadan,
Michael Holst
Publication year - 2022
Publication title -
contemporary mathematics
Language(s) - English
Resource type - Journals
eISSN - 2705-1064
pISSN - 2705-1056
DOI - 10.37256/cm.3220221278
Subject(s) - mathematics , pure mathematics , metric (unit) , laplace operator , covariant derivative , differential operator , levi civita connection , sobolev space , einstein , mathematical analysis , riemannian geometry , fundamental theorem of riemannian geometry , ricci curvature , curvature , geometry , mathematical physics , operations management , economics
The study of Einstein constraint equations in general relativity naturally leads to considering Riemannian manifolds equipped with nonsmooth metrics. There are several important differential operators on Riemannian manifolds whose definitions depend on the metric: gradient, divergence, Laplacian, covariant derivative, conformal Killing operator, and vector Laplacian, among others. In this article, we study the approximation of such operators, defined using a rough metric, by the corresponding operators defined using a smooth metric. This paves the road to understanding to what extent the nice properties such operators possess, when defined with smooth metric, will transfer over to the corresponding operators defined using a nonsmooth metric. These properties are often assumed to hold when working with rough metrics, but to date the supporting literature is slim.