Curvature Generates Geometry: From Spaces of Constant Curvature to Comparison Yardsticks

This text uses the unified trigonometric functions sinₖ,cosₖ,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, the metric of a two-dimensional complete simply connected space of constant curvature is uniquely generated in geodesic polar coordinates by sinₖ,dsₖ²=dr²+sinₖ²(r)\,dθ²,where sinₖ is not an ad hoc auxiliary function but the necessary solution of the constant-curvature condition hᵣᵣ+kh=0 together with the initial conditions $h(0)=0$, hᵣ(0)=1. Curvature k generates sinₖ through this ODE, sinₖ 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≥3. The text unfolds in three levels. **The first level (Chapters 1–6)** fixes the curvature k and traces the manifestations of sinₖ 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 ε⁴ coefficient βₙ=-π²/24(n⁴+14n²-9), two-mode cross term γm,n=-π²/3(m²n²+2m²+2n²-3)), asymptotic behavior as k→±∞, 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 sinₖ,cosₖ,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≥3, the geometric interpretation of the multi-mode ε³ coefficient, etc.).