The Geometry of Dilation- and Shear-Deformed Spaces

Abstract

This paper develops a deformation-field geometry for spaces whose local frames may undergo internal dilation, compression, and shear. The basic datum is an admissible dilation-shear field $P$ over a selected metric-compatible reference geometry $(M,\bar g,\bar\nabla)$. It represents the induced metric by $g=P^T\bar gP$ and compares tangent data through the reference representative $\bar V=PV$. The covariant derivative associated with the natural connection is defined by \[ \nabla_XV=P^{-1}\bar\nabla_X(PV), \] with local connection coefficients \[ \Gamma=\Lambda=P^{-1}\bar\Gamma P+P^{-1}dP . \] Thus the total dilation-shear compensation is represented by the natural connection coefficients $\Gamma=\Lambda$. If $\bar\nabla\bar g=0$, then $\nabla g=0$; hence the covariant derivative associated with the natural connection has zero nonmetricity. The general distinction from Levi-Civita geometry lies instead in torsion and in the deformation origin of the comparison. The Levi-Civita connection coefficients $\mathring\Gamma[g]$ are retained as the torsion-free metric connection of the induced metric layer; they appear naturally in fully isometric realizations, but they are not an additional term to be added to the natural connection. The same pullback rule covers composite references that already contain both an isometric realization and an internal dilation-shear natural connection. Examples involving one-dimensional dilation, conformal deformation, anisotropic dilation, shear, and spherical geometries distinguish metric representation, natural dilation-shear comparison, Levi-Civita comparison, and embedded realization.