Curvature Generates Geometry: Unified Formulas and Systematic Derivations for Spherical, Euclidean, and Hyperbolic Geometries

This paper establishes a complete unified formula system from the second-order linear ordinary differential equation $y''+ky=0$ to constant-curvature geometry, comparison geometry, and rigidity theorems, with the framework of **curvature-generated geometry** (i.e., curvature generates geometry) as the organizing thread. The core generating chain is as follows: curvature $k$ generates the unified trigonometric functions $\sin_k,\cos_k$ through the ODE; $\sin_k$ generates the constant-curvature metric $g_k=dr^2+\sin_k^2(r)\,d\theta^2$ in geodesic polar coordinates; the metric then generates all downstream geometric quantities, including Jacobi fields, the exponential map, the Hessian and Laplacian of the distance function, volume, trigonometry, and circle geometry. The paper takes the complex parameter $\alpha$ (with $k=-\alpha^2$) and the unified Euler formula $e^{\alpha r}=\cos_k(r)+\alpha\sin_k(r)$ as the algebraic anchor, unifying spherical, Euclidean, and hyperbolic geometries as the imaginary, zero, and real branches of $\alpha$, respectively, and provides chapter-by-chapter systematic derivations in the form of “unified form + three-geometry explicit forms + comparison tables.” The 13 chapters form a complete derivation chain from the ODE to comparison rigidity: unified trigonometric functions and generating geometry; the full set of elementary functions; Christoffel symbols and the curvature tensor; Jacobi fields and conjugate points; the exponential map and pullback metric; distance Hessian, Laplacian, and convexity; volume, tubular neighborhoods, and Steiner formulas; trigonometry, spherical harmonics, and complex trigonometry; circle geometry and the isoperimetric inequality; small-curvature expansions and $k$-responses; the five comparison theorems of Rauch, Hessian, Laplacian, Bishop–Gromov, and Toponogov; rigidity theorems at equality in comparison and quantitative stability; formula tables, dependency trees, and open problems. Appendices provide three-geometry comparison tables, a complex-parameter manual, a symbol index, and a Euclidean-limit operation manual. The core contributions of this paper are: **taking the generating chain as the principal thread, systematically linking all core formulas of constant-curvature geometry and comparison geometry into a directed, derivable, and self-contained system**; **using the unified Euler formula as an algebraic anchor to unify the Pythagorean identity, addition formulas, power series, Jacobi fields, the exponential map, and model angles**; and **adopting a methodology of item-by-item three-geometry realization, making the correspondences among the three geometries systematic and visible**. The paper provides complete proofs, or proof ideas with key steps, for each core formula; the proofs of the key comparison and rigidity theorems are closed within the paper (the shape-operator Riccati equation for Hessian/Laplacian is inlined in Chapter 11, Bishop–Gromov rigidity relies on Step 5 of the Chapter 11 comparison, and the two-dimensional case of constant-curvature global rigidity is self-contained via the covering-map route). The paper can serve as a systematic reference for constant-curvature geometry and comparison geometry; its value lies in the systematic organization of unified formulas, the item-by-item derivation of three-geometry formulas, and the establishment of a complete derivation chain from the ODE to comparison rigidity.