Linear Strain Tensors on Hyperbolic Surfaces and Asymptotic Theories for Thin Shells

We perform a detailed analysis of the solvability of linear strain equations on hyperbolic surfaces. We prove that if the surface is a smooth noncharacteristic region, any first order infinitesimal isometry can be matched to an infinitesimal isometry of an arbitrarily high order. The implications of this result for the elasticity of thin hyperbolic shells are discussed.