Abstract Shear transformations are fundamental in modeling deformations in flat spaces, yet classical formulations fail on curved manifolds where curvature and topology impose intrinsic constraints. Many real-world systems, such as planetary atmospheres, elastic shells, and biological membranes, exhibit deformations in non-Euclidean settings where traditional shear lacks a natural definition. This work develops a unified, coordinate-invariant framework for shear-like deformations on Riemannian and Lorentzian manifolds. Starting with a vector-field-driven shear on the two-dimensional sphere, we analyze its geometry using the Jacobian and extend this concept by employing a tensorial strain model derived from covariant derivatives and Lie derivatives of the metric. The resulting formulation reveals fundamental links between strain evolution, parallel transport, and holonomy. Applications span multiple domains: (i) in General Relativity, we establish how strain evolution couples to the Riemann tensor, connecting tidal forces, geodesic deviation, and gravitational-wave memory effects; (ii) in Elastic Membranes, we compute explicit strain components for a shear field on S2 and verify compatibility conditions under curvature and torsion, exposing geometric frustration; and (iii) in Atmospheric and Geological Flows, we employ the deformation gradient, viola transform, and three-dimensional Einstein tensor to model curvature-induced stresses and large-scale flow dynamics. This framework bridges differential geometry, continuum mechanics, and physical modeling, providing new tools for understanding deformation in curved spaces and offering predictive insights for both astrophysical and geophysical systems.
A. Ahmed (Wed,) studied this question.