Geometry Generated by Complex-Parameter Curvature: A Unified Generation Chain for the Spherical, Euclidean, and Hyperbolic Three Sections

This paper takes the complex parameter κ and the curvature k=-κ² as the sole parameter, and establishes a generation chain starting from the constant-curvature Jacobi equation. It unifies **spherical geometry** (κ=iμ), **Euclidean geometry** (κ=0), and **hyperbolic geometry** (κ=λ₀>0) as the realization of a single complex-parameter formula on three sections of the κ-plane:κ→sin_κ,cos_κ→ g_κ=dr²+sin_κ²(r)dΩₙ₋₁²→ fields, exponential map, distance, volume, spectrum, trigonometry, circles, comparison, rigidity\.The unified trigonometric functions sin_κ=(κ r)/κ and cos_κ=(κ r) are uniquely determined by the initial value problem for J''-κ²J=0, and the unified Euler formula eκ r=cos_κ+κsin_κ yields the Pythagorean identity, derivative relations, and addition formulas. The second link of the generation chain uniquely produces g_κ from the constant-curvature condition hᵣᵣ-κ²h=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 Sⁿ(1/μ), Euclidean space Rⁿ, and hyperbolic space Hⁿ(1/λ₀). Continuing along the chain, the paper uniformly treats normal Jacobi fields J=cos_κ u+sin_κ v, matrix Jacobi fields and the Riccati equation, conjugate points rₘ=mπ i/κ, real/complex Morse indices and the Stokes jump, the exponential map expₚ(v)=cos_κ(r)p+sin_κ(r)v/r, the pullback metric, the logarithmic map, the Hessian of the distance function Hessρ=_κ(ρ)(g-dρ²), and the Laplacian of the distance function Δρ=(n-1)_κ(ρ); 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_κ(x)=κ/2κ x/2-1x, proves that its generating function is the curvature scaling of ζ(2m) and that its Mellin transform is κ¹⁻ˢΓ(s)ζ(s), and strictly distinguishes the spectral zeta from this Mellin transform. Chapter 8 discusses almost complex structures and holomorphic geometry on H_κ, establishes that the case $n=2$ is automatically Kähler and that for n≥3 the space is conformally flat but not Kähler, and lists the compatibility of almost complex structures for general complex κ 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 κ on three sections**. For real or purely imaginary κ, H_κ is precisely the classical constant-curvature space form: sphere, Euclidean space, or hyperbolic space. For general complex κ, 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 ζ from geometry.”