Geometry Generated by Complex-Parameter Curvature: A Unified Generation Chain for the Spherical, Euclidean, and Hyperbolic Three Sections
Year: 2026
Type: preprint
Abstract: This paper takes the complex parameter $\kappa\in\mathbb{C}$ and the curvature $k=-\kappa^2$ as the sole parameter, and establishes a generation chain starting from the constant-curvature Jacobi equation. It unifies **spherical geometry** ($\kappa=i\mu$), **Euclidean geometry** ($\kappa=0$), and **hyperbolic geometry** ($\kappa=\lambda_0>0$) as the realization of a single complex-parameter formula on three sections of the $\kappa$-plane:$$\kappa\to\sin_\kappa,\cos_\kappa\to g_\kappa=dr^2+\sin_\kappa^2(r)d\Omega_{n-1}^2\to\{\text{Jacobi fields, exponential map, distance, volume, spectrum, trigonometry, circles, comparison, rigidity}\}.$$The unified trigonometric functions $\sin_\kappa=\sinh(\kappa r)/\kappa$ and $\cos_\kappa=\cosh(\kappa r)$ are uniquely determined by the initial value problem for $J''-\kappa^2J=0$, and the unified Euler formula $e^{\kappa r}=\cos_\kappa+\kappa\sin_\kappa$ yields the Pythagorean identity, derivative relations, and addition formulas. The second link of the generation chain uniquely produces $g_\kappa$ from the constant-curvature condition $h_{rr}-\kappa^2h=0$ together with the standard geodesic polar-coordinate initial values; its Christoffel symbols, curvature tensor, and Ricci curvature are obtained by direct computation without relying on embeddings. The three sections degenerate respectively to the classical constant-curvature metrics on the sphere $S^n(1/\mu)$, Euclidean space $\mathbb{R}^n$, and hyperbolic space $\mathbb{H}^n(1/\lambda_0)$. Continuing along the chain, the paper uniformly treats normal Jacobi fields $J=\cos_\kappa u+\sin_\kappa v$, matrix Jacobi fields and the Riccati equation, conjugate points $r_m=m\pi i/\kappa$, real/complex Morse indices and the Stokes jump, the exponential map $\exp_p(v)=\cos_\kappa(r)p+\sin_\kappa(r)v/r$, the pullback metric, the logarithmic map, the Hessian of the distance function $\operatorname{Hess}\rho=\cot_\kappa(\rho)(g-d\rho^2)$, and the Laplacian of the distance function $\Delta\rho=(n-1)\cot_\kappa(\rho)$; it also gives the volume element, the discrete spectrum on the sphere and the continuous spectrum on hyperbolic space, and heat-kernel and heat-trace asymptotics. Chapter 9 derives, from the unified embedding, the complete trigonometry, circumference and area of circles, the disk quadratic relation, and the isoperimetric inequality for the spherical, Euclidean, and hyperbolic three sections. Chapter 10 takes Riccati matrix comparison as its core and establishes five comparison theorems—Rauch, Hessian, Laplacian, Bishop–Gromov, and Toponogov—together with their rigidity, uniformly formulating the comparison conclusions for the three model spaces: sphere, Euclidean space, and hyperbolic space. Chapter 7 defines the curvature function $F_\kappa(x)=\frac{\kappa}{2}\coth\frac{\kappa x}{2}-\frac1x$, proves that its generating function is the curvature scaling of $\zeta(2m)$ and that its Mellin transform is $\kappa^{1-s}\Gamma(s)\zeta(s)$, and strictly distinguishes the spectral zeta from this Mellin transform. Chapter 8 discusses almost complex structures and holomorphic geometry on $\mathbb{H}_\kappa$, establishes that the case $n=2$ is automatically Kähler and that for $n\ge3$ the space is conformally flat but not Kähler, and lists the compatibility of almost complex structures for general complex $\kappa$ as an open problem. The core of this paper is that **spherical, Euclidean, and hyperbolic geometry are not three independent theories, but item-by-item realizations, along the generation chain, of a single complex parameter $\kappa$ on three sections**. For real or purely imaginary $\kappa$, $\mathbb{H}_\kappa$ is precisely the classical constant-curvature space form: sphere, Euclidean space, or hyperbolic space. For general complex $\kappa$, the metric is a complex-valued quadratic form, and the relevant formulas hold as analytic continuation identities within domains of convergence, branch conventions, and non-degenerate regions. Throughout, conclusions are labeled with a four-level reliability scheme A/B/C/D and accompanied by a list of discouraged claims, explicitly excluding overclaims such as “proving RH” or “decoding the zeros of $\zeta$ from geometry.”
Author: NI, Chuangao
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225640086
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