Lie Sphere Geometry

Abstract

This chapter presents the method of moving frames in Lie sphere geometry. This involves a number of new ideas, beginning with the fact that some Lie sphere transformations are not diffeomorphisms of space S ³, but rather of the unit tangent bundle of S ³. This we identify with the set of pencils of oriented spheres in S ³, which is identified with the set $\varLambda$ of all lines in the quadric hypersurface $Q \subset \mathbf{P}(\mathbf{R}^{4,2})$. The set $\varLambda$ is a five-dimensional subspace of the Grassmannian G(2, 6). The Lie sphere transformations are the projective transformations of P(R ^4, 2) that send Q to Q. This is a Lie group acting transitively on $\varLambda$. The Lie sphere transformations taking points of S ³ to points of S ³ are exactly the Möbius transformations, which form a proper subgroup of the Lie sphere group. In particular, the isometry groups of the space forms are natural subgroups of the Lie sphere group. There is a contact structure on $\varLambda$ invariant under the Lie sphere group. A surface immersed in a space form with a unit normal vector field has an equivariant Legendre lift into $\varLambda$. A surface conformally immersed into Möbius space with an oriented tangent sphere map has an equivariant Legendre lift into $\varLambda$. This chapter studies Legendre immersions of surfaces into this homogeneous space $\varLambda$ under the action of the Lie sphere group. A major application is a proof that all Dupin immersions of surfaces in a space form are Lie sphere congruent to each other.