Reshetnyak rigidity for Riemannian manifolds

Abstract

We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map $f:M\to N$ between two oriented Riemannian manifolds, whose differential is almost everywhere an orientation-preserving isometry, is an isometric immersion. This theorem was previously proved using regularity theory for conformal maps; we give a new, simple proof, by generalizing the Piola identity for the cofactor operator. Second, we prove that if there exists a sequence of mapping $f_n:M\to N$, whose differentials converge in $L^p$ to the set of orientation-preserving isometries, then there exists a subsequence converging to an isometric immersion. These results are generalizations of celebrated rigidity theorems by Liouville (1850) and Reshetnyak (1967) from Euclidean to Riemannian settings. Finally, we describe applications of these theorems to non-Euclidean elasticity and to convergence notions of manifolds.

Formerly titled: Asymptotic rigidity of Riemannian manifolds. Version 2: a completely rewritten Section 2 (Section 3 in version 1) with new proofs; major edits in Sections 1 and 3.1 (formerly 2.2) and the appendices. Version 3: Title change; Section 1.3 (applications) expanded; other minor changes

References in corpus (4)

• The emergence of torsion in the continuum limit of distributed edge-dislocations
• Riemannian surfaces with torsion as homogenization limits of locally-Euclidean surfaces with dislocation-type singularities
• Intrinsic weak derivatives and Sobolev spaces between manifolds
• Limits of elastic models of converging Riemannian manifolds

Cited by in corpus (5)

• On Asymptotic Rigidity and Continuity Problems in Nonlinear Elasticity on Manifolds and Hypersurfaces
• Asymptotic rigidity for shells in non-Euclidean elasticity
• Stability of isometric immersions of hypersurfaces
• Linearization in incompatible elasticity for general ambient spaces
• From Volterra dislocations to strain-gradient plasticity