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 $g=P^T\bar gP$. 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.