General Deformation Geometry

We develop a general deformation geometry in which the primitive datum is an invertible deformation field \(A\) relative to a selected reference geometry. Instead of prescribing only a metric, the deformation field determines both the induced metric $g(U,V)=\bar g(AU,AV)$ and the transported connection form $ \Gamma=A^{-1}\bar\Gamma A+A^{-1}\mathrm dA$. Thus the induced metric and the transported connection are determined by one deformation field $A$, through the reference relation $\bar V=AV$. For an admissible factorization $A=OP$, the dilation-shear factor $P$ changes the metric, while the $\bar g$-orthogonal rotation-twist factor $O$ leaves the metric unchanged but changes the connection. The connection formula gives the coupling term $P^{-1}O^{-1}\mathrm dO\,P$, showing that rotation-twist and dilation-shear do not simply add as independent effects. The theory also distinguishes metric realization from deformation-field realization: a full isometric realization selects the Levi-Civita connection of the realized metric, whereas a general deformation field may produce a residual distortion relative to that Levi-Civita connection. In this way the framework extends Riemannian and pseudo-Riemannian metric geometry to a deformation-induced metric-connection geometry, while differing from metric-affine geometry by tying the metric and connection to the same deformation field and reference connection.