Transformation Groups and Erlangen Program - Main Theorem

TheoremClassification of Geometries of Constant Curvature

Up to isomorphism, there are exactly three simply-connected complete Riemannian manifolds of constant curvature in each dimension $n$ :

Positive curvature $K = +1$ : Sphere $S^n$
Zero curvature $K = 0$ : Euclidean space $\mathbb{R}^n$
Negative curvature $K = -1$ : Hyperbolic space $\mathbb{H}^n$

These correspond to elliptic, parabolic, and hyperbolic geometries in Klein's classification.

This theorem unifies geometric types by curvature, showing the three classical geometries exhaust constant-curvature possibilities. Their isometry groups are $SO(n+1)$ , $E(n)$ , and $SO^+(n,1)$ respectively.