The Geometry of Dilation- and Shear-Deformed Spaces

This paper develops a deformation-field geometry for spaces whose local frames may undergo internal stretching, compression, and shear. Ordinary Riemannian geometry takes an intrinsic metric geometry \ ( (M, g) \) as the given datum and uses its Levi-Civita comparison. The present framework retains additional data: a fixed reference metric geometry and a deformation field \ (P\) representing \ (g\) by \ (g=PT gP\). This makes the dilation--shear structure relative to the fixed reference visible. The deformation field yields a dilation--shear compensation \ (=P^-1 P\), and the natural total comparison connection is \ (=+\), where \ (\) is the Levi-Civita connection of the represented metric. Curvature, torsion, and nonmetricity of \ (\) are then determined by \ (\) and \ (\), rather than postulated as independent affine data. Examples involving one-dimensional stretching, conformal deformation, anisotropic dilation, shear, and spherical geometries distinguish metric curvature, embedded realization, and internal deformation non-uniformity. Status: Submitted manuscript. This is the author-created version submitted to the Journal of Differential Geometry.