  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 approach each geometry is a (connected) manifold endowed with a group of automorphisms, that is, a Lie group \(G\) of  “motions” that acts transitively on the  manifold, such that two figures are regarded as congruent if and only if there exists an element of the appropriate Lie group \(G\) that transforms one of the figures into the other. A generalized geometry  in Klein’s sense shifts the emphasis from the underlying manifold or  space to the group. Thus a Klein geometry (space) consists of, (1) a  smooth manifold, (2) a Lie group \(G\) (the principal group of the geometry), and (3) a transitive action of \(G\) on the manifold.  Besides being “global”, a Klein geometry (space) is  completely homogeneous in the sense that its points cannot be  distinguished on the basis of geometric relations because the  transitive group action preserves such relations. 