Möbius Geometry

Abstract

This chapter introduces conformal geometry and Liouville’s characterization of conformal transformations of Euclidean space. Through stereographic projection these are all globally defined conformal transformations of the sphere S ³. The Möbius group $\mathbf{M\ddot{o}b}$ is the group of all conformal transformations of S ³. It is a ten dimensional Lie group containing the group of isometries of each of the space forms as a subgroup. Möbius space $\mathcal{M}$ is the homogeneous space consisting of the sphere S ³ acted upon by $\mathbf{M\ddot{o}b}$. $\mathcal{M}$ has a conformal structure invariant under the action of $\mathbf{M\ddot{o}b}$. The reduction procedure is applied to Möbius frames. The space forms are each equivariantly embedded into Möbius geometry. Conformally invariant properties, such as Willmore immersion, or isothermic immersion, or Dupin immersion, have characterizations in terms of the Möbius invariants. Oriented spheres in Möbius space provide the appropriate geometric interpretation of the vectors of a frame field.