The Geometry of Dilation- and Shear-Deformed Spaces

TL;DR

This paper introduces a deformation-field geometry framework for spaces with local stretching, compression, and shear, extending Riemannian geometry by incorporating a fixed reference metric and deformation field.

Contribution

It develops a new geometric approach that explicitly models internal deformation data, revealing dilation-shear structures relative to a fixed reference metric.

Findings

Deformation field yields dilation-shear compensation 1

Total comparison connection combines Levi-Civita connection and deformation 1

Examples distinguish metric curvature, realization, and deformation non-uniformity

Abstract

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 \(g=P^T\bar gP\). This makes the dilation-shear structure relative to the fixed reference visible. The deformation field yields a dilation-shear compensation \(\Lambda=P^{-1}\bar\nabla P\), and the natural total comparison connection is \(\Gamma=\mathring\Gamma+\Lambda\), where \(\mathring\Gamma\) is the Levi-Civita connection of the represented metric. Curvature, torsion, and nonmetricity of \(\Gamma\) are then determined by \(\mathring\Gamma\) and \(\Lambda\), 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.