Relativistic Deformation of Geometry through Function C(v): Scalar Deformation Flow and the Geometric Classification of 3-Manifolds

Abstract: We introduce the scalar deformation function C(v), which captures how local geometric structures respond to motion at velocity v. This function exhibits smooth analytic behavior and defines a critical velocity vc beyond which the geometry compresses. Extending C(v) into a flow C(v, tau), we construct a scalar analogue of Ricci flow that governs the evolution of geometric configurations toward symmetric, stable states without singularities. The flow is derived from a variational energy functional and satisfies global existence and convergence properties. We show that this scalar evolution provides a pathway for topological classification of three-manifolds through conformal smoothing and energy minimization, offering a curvature-free geometric mechanism rooted in analytic deformation. The resulting framework combines techniques from differential geometry and dynamical systems and may serve as a minimal geometric model for structure formation in relativistic contexts.