form where , , is an orthogonal matrix, and is 0 or 2. The group of Möbius transformations is also called the Möbius group. The orientation-preserving Möbius transformations form the connected component of the identity in the Möbius group. In dimension , the orientation-preserving Möbius transformations are exactly the maps of the Riemann sphere covered here. The orientation-reversing ones are obtained from these by complex conjugation. The domain of Möbius transformations, i.e. , is homeomorphic to the n-dimensional sphere . The canonical isomorphism between these two spaces is the Cayley transform, which is itself a Möbius transformation of . This identification means that Möbius transformations can also be thought of as conformal isomorphisms of . The n-sphere, together with action of the Möbius group, is a geometric structure (in the sense of Klein's Erlangen program) called Möbius geometry. Applications Lorentz transformation An isomorphism of the Möbius group with the Lorentz group was noted by several authors: Based on previous work of Felix Klein (1893, 1897) on automorphic functions related to hyperbolic geometry and Möbius geometry, Gustav Herglotz (1909) showed that hyperbolic motions (i.e. isometric automorphisms of a hyperbolic space) transforming the unit sphere into itself correspond to Lorentz transformations, by which Herglotz was able to classify the one-parameter Lorentz transformations into loxodromic, elliptic, hyperbolic, and parabolic groups. Other authors include Emil Artin (1957), H. S. M. Coxeter (1965), and Roger Penrose, Wolfgang Rindler (1984), Tristan Needham (1997) and W. M. Olivia (2002). Minkowski space consists of the four-dimensional real coordinate space R4 consisting of the space of ordered quadruples (x0,x1,x2,x3) of real numbers, together with a quadratic form Borrowing terminology from special relativity, points with are considered timelike; in addition, if , then the point is called future-pointing. Points with are called spacelike. The null cone S consists of those points where ; the future null cone N+ are those points on the null cone with . The celestial sphere is then identified with the collection of rays in N+ whose initial point is the origin of R4. The collection of linear transformations on R4 with positive determinant preserving the quadratic form Q and preserving the time direction form the restricted Lorentz group SO+(1,3). In connection with the geometry of the celestial sphere, the group of transformations SO+(1,3) is identified with the group PSL(2,C) of Möbius transformations of the sphere. To each , associate the hermitian matrix The determinant of