Curvature-Driven Shear Deformation: A Unified Geometric Framework for Gravitational Memory and Geophysical Flows

This work addresses a fundamental question: How does intrinsic curvature alter the definition and evolution of shear deformation in curved geometries? Classical approaches based on flat-space assumptions fail due to coordinate dependence, path-dependent parallelism (holonomy), and topological constraints, creating significant gaps in understanding systems ranging from relativistic spacetimes to planetary atmospheres. We resolve this by developing a unified geometric framework that intrinsically defines shear deforma tion on Riemannian and Lorentzian manifolds. Using coordinate-invariant methods—including the Lie derivative of the metric, covariant derivatives, and Jacobi fields—we construct a curvature-compatible strain tensor and derive its evolution equation directly coupled to the Riemann curvature tensor. Our approach yields three transformative outcomes: First, in General Relativity, we unify tidal forces, geodesic deviation, and gravitational wave memory through explicit spacetime strain-curvature coupling. Second, for elastic membranes on spheres, we quantify geometric frustration caused by topological obstruc tion, explaining anomalous stress relaxation. Third, in geophysical flows, we predict curvature-induced instabilities like jet stream localization using the Einstein tensor, revealing phenomena absent in flat-space models. By synthesizing differential geometry, continuum mechanics, and relativity, this work establishes fun damental tools for deformation analysis in curved spaces, with direct applications to gravitational wave detection, atmospheric science, and soft matter engineering.