2016/07/31 by Gui‐Qiang Chen, Gui-Qiang G. Chen, Siran Li · 21 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Curl (programming language) #Curvature of Riemannian manifolds #Geometric Analysis and Curvature Flows #Geometry #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Physics #Pure mathematics #Ricci curvature #Ricci-flat manifold #Riemannian geometry #Riemannian manifold #Rigidity (electromagnetism) #Scalar curvature #Sectional curvature #math.AP #math.DG #math.FA #math.MG #msc:35B35 #msc:35M30 #msc:53C21 #msc:53C24 #msc:53C42 #msc:53C45 #msc:57R40 #msc:57R42 #msc:58A05 #msc:58A14 #msc:58A15 #msc:58A17 #msc:58J10. #msc:58K30 #msc:58Z05
paper · pdf · doi:10.1007/s12220-017-9893-1
published in Journal of Geometric Analysis 28(3), 1957-2007 (Springer Science+Business Media) · 42 pages, Journal of Geometric Analysis, 2017
openalex publication_date 2017/08/18 · arxiv created 2017/08/26 · arxiv updated 2017/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 corresponding different metrics.