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 S3, but rather of the unit tangent bundle of S3. This we identify with the set of pencils of oriented spheres in S3, 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(R4, 2) that send Q to Q. This is a Lie group acting transitively on \( \varLambda \). The Lie sphere transformations taking points of S3 to points of S3 are exactly the Mobius 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 Mobius 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.