The Complex-Scaling Parameter Family: Unifying Euclidean, Spherical, and Hyperbolic Geometries via Limits

We construct and systematically analyze a family of metrics generated by the complex scaling parameter $\mathcal{K}=\kappa+i\lambda$, $$ ds^2=du^2+e^{2\kappa u}\cos^2(\lambda u)\,d\theta^2, $$ together with its higher-dimensional generalizations. The generator $\Phi(u)=e^{\kappa u}\cos(\lambda u)$ satisfies a second-order linear equation with constant coefficients; via a gauge transformation, this equation is equivalent to the Jacobi field equation of a constant-curvature space. The metric is periodically degenerate at the zeros $u_n=(2n+1)\pi/(2\lambda)$: when $\kappa\neq0$, the curvature diverges as a first-order pole, whereas when $\kappa=0$, the curvature remains finite but conical singularities or smooth poles occur. We compute the connection, Gaussian curvature, Ricci tensor, and scalar curvature; analyze the cone-angle structure at the zeros; and prove that open cells are geodesically incomplete, that non-radial geodesics are trapped by a potential barrier, and that radial Jacobi fields are bounded on finite intervals, with the pole being a simple conjugate point. At the measure level, we prove that the circumference tends to zero while the area remains finite, and that the area ratio is monotonically decreasing; we also verify the curvature integral by means of a cell-by-cell Gauss–Bonnet-type consistency relation. Topologically, we distinguish three constructions: the disjoint union of open cells, the quotient-topological sphere chain, and the metric completion; we show that the sphere chain is the universal covering of a pinched sphere. In the variational part, we prove that minimal surfaces are confined to a single cell and establish the finiteness of the curvature integral. We prove that along three parameter-limit paths, the metric family converges, in the local $C^\infty$ geometry and in the pointed Gromov–Hausdorff sense, to a hyperbolic cylinder, a conical sphere, and a flat cylinder, respectively; global homeomorphic convergence fails, and the topology undergoes a phase transition in the limit. Thus, the three classical constant-curvature geometries are unified as local geometric limits of this parameter family on the boundary of parameter space.