Global Weak Rigidity of the Gauss–Codazzi–Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity
| Autor: | Siran Li, Gui-Qiang Chen |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
58A17 58A15 53C42 58A14 53C45 35B35 01 natural sciences Riemannian manifolds Isometric embedding Rigidity (electromagnetism) Lower regularity Mathematics Weak convergence Metric Geometry (math.MG) Global 16. Peace & justice Functional Analysis (math.FA) Mathematics - Functional Analysis Weak rigidity Geometric div-curl lemma symbols Riemann curvature tensor 010307 mathematical physics Mathematics::Differential Geometry div-curl structure Analysis of PDEs (math.AP) Mathematics - Differential Geometry 58Z05 Gauss–Codazzi–Ricci equations Isometric immersion Banach space Cartan formalism 35M30 53C21 Approximate solutions Secondary: 57R40 Article 58J10 58K30 symbols.namesake Primary: 53C24 Mathematics - Analysis of PDEs Mathematics - Metric Geometry 0103 physical sciences Compactness theorem Euclidean geometry FOS: Mathematics 0101 mathematics Primary: 53C24 53C42 53C21 53C45 57R42 35M30 35B35 58A15 58J10. Secondary: 57R40 58A14 58A17 58A05 58K30 58Z05 010102 general mathematics 58A05 Differential geometry Differential Geometry (math.DG) Intrinsic 57R42 Geometry and Topology |
| Zdroj: | Journal of Geometric Analysis |
| ISSN: | 1559-002X 1050-6926 |
| Popis: | We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates, and provide further insights of the previous local results and arguments. The critical case has also been analyzed. To achieve this, we first reformulate the GCR equations with div-curl structure intrinsically on Riemannian manifolds and develop a global, intrinsic version of the div-curl lemma and other nonlinear techniques to tackle the global weak rigidity on manifolds. In particular, a general functional-analytic compensated compactness theorem on Banach spaces has been established, which includes the intrinsic div-curl lemma on Riemannian manifolds as a special case. The equivalence of global isometric immersions, the Cartan formalism, and the GCR equations on the Riemannian manifolds with lower regularity is established. We also prove a new weak rigidity result along the way, pertaining to the Cartan formalism, for Riemannian manifolds with lower regularity, and extend the weak rigidity results for Riemannian manifolds with unfixed metrics. 42 pages, Journal of Geometric Analysis, 2017 |
| Databáze: | OpenAIRE |
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