The unit sphere and CR geometry

Abstract

CR geometry considers the interplay between real and complex spaces. The name itself has an interesting history, which we do not discuss here, other than to say that CR stands both for Cauchy-Riemann and for Complex-Real. See [DT] for a survey of CR Geometry and its connections with other branches of mathematics. See [J] for a good exposition of the theory of CR structures, and see [BER] for a definitive treatment of CR mappings.

In this chapter we consider simple aspects of the CR geometry of the unit sphere in $\mathbf{C}^n$ and relate them to the holomorphic automorphism group of the unit ball. The unit sphere $S^{2n-1}$ in $\mathbf{R}^{2n}$ becomes an object in CR Geometry after we identify $\mathbf{R}^{2n}$ with $\mathbf{C}^n$. Given this identification, we discover that the tangent directions to the sphere do not all behave the same way from the point of view of complex analysis. This issue lies at the foundation of CR Geometry and we will develop it in Section 1.