Curvature Generates Geometry: From Spaces of Constant Curvature to Comparison Yardsticks
Year: 2026
Type: preprint
Abstract: This text uses the unified trigonometric functions $\sin_k,\cos_k,\cot_k$ as a guiding thread to systematically trace how the ordinary differential equation $y''+ky=0$ manifests itself at various levels of Riemannian geometry. The central thesis of the book is: for a given curvature $k\in\mathbb{R}$, the metric of a two-dimensional complete simply connected space of constant curvature is uniquely generated in geodesic polar coordinates by $\sin_k$,$$ds_k^2=dr^2+\sin_k^2(r)\,d\theta^2,$$where $\sin_k$ is not an ad hoc auxiliary function but the necessary solution of the constant-curvature condition $h_{rr}+kh=0$ together with the initial conditions $h(0)=0$, $h_r(0)=1$. Curvature $k$ generates $\sin_k$ through this ODE, $\sin_k$ generates the geometry through the metric, and the geometric invariants (distance, Jacobi fields, Hessian, volume, trigonometry, circle geometry) are uniquely determined by the metric. This “generation chain” is fully self-contained in dimension two (once global uniqueness for $k>0$ is completed via smoothness at the pole and radial minimality, no Killing–Hopf classification is needed); the classification theorem for space forms is invoked only for $n\ge3$. The text unfolds in three levels. **The first level (Chapters 1–6)** fixes the curvature $k$ and traces the manifestations of $\sin_k$ in geodesic variations (Jacobi fields), the distance function (Hessian/Laplacian), volume (Bishop–Gromov), trigonometry (cosine/sine/tangent laws and angle sum), and circle geometry (circumference/area/isoperimetric inequality). **The second level (Chapter 7)** turns to a parametric study: how the various geometric quantities respond when $k$ varies continuously—small-curvature expansions, first-order corrections to the third side and interior angles, monotonicity of circumference/area, the $k$-expansion of the isoperimetric deficit (single-mode $\varepsilon^4$ coefficient $\beta_n=-\frac{\pi^2}{24}(n^4+14n^2-9)$, two-mode cross term $\gamma_{m,n}=-\frac{\pi^2}{3}(m^2n^2+2m^2+2n^2-3)$), asymptotic behavior as $k\to\pm\infty$, and critical phenomena for $k>0$. **The third level (Chapter 8)** completes the role reversal: spaces of constant curvature are no longer the objects under study but rather comparison yardsticks—the model functions in the four comparison theorems of Toponogov, Rauch, Hessian/Laplacian, and Bishop–Gromov are precisely the same family $\sin_k,\cos_k,\cot_k$ appearing in the exact formulas of the preceding seven chapters. The methodology of the text consists in: (1) treating $k$ as a free parameter to unify the spherical, Euclidean, and hyperbolic cases, avoiding separate treatments of three sets of formulas; (2) giving two or three proofs for each geometric quantity (coordinate computation, Jacobi-field method, embedding method, Riccati approach) for cross-validation; (3) clearly distinguishing self-contained parts from cited parts, and giving an explicit embedding proof of global uniqueness in dimension two for $k>0$ that does not rely on Killing–Hopf; (4) flagging open problems (the curvature symmetry conjecture for $n\ge3$, the geometric interpretation of the multi-mode $\varepsilon^3$ coefficient, etc.).
Author: NI, Chuangao
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225251919