Klein's Erlanger program provides a unified framework by associating each geometry with a Lie group of automorphisms acting transitively on a manifold.

In his Erlanger program, Klein provided a unified approach to the various “global” geometries by showing that each of the geometries is characterized by a particular group of transformations: Euclidean geometry is characterized by the group of translations and rotations in the plane; the geometry of the sphere \(S^{2}\) is characterized by the orthogonal group \(O(3)\); and the geometry of the hyperbolic plane is characterized by the pseudo-orthogonal group \(O(1, 2)\). In Klein’s approac