CR Functions and Maps

In this chapter, we present the definitions and basic properties of CR functions and CR maps. CR functions are analogous to holomorphic functions on a complex manifold. However, there are’ important differences. For example, CR functions are not always smooth. There are relationships between CR functions on an imbedded CR manifold and holomorphic functions on the ambient ℂ ⁿ . For example, the restriction of a holomorphic function to a CR submanifold is a CR function. However, CR functions do not always extend to holomorphic functions. In this chapter, we show that real analytic CR functions on a real analytic CR submanifold locally extend to holomorphic functions. A C ^∞ version of this is also given. The chapter ends with a discussion of CR maps between CR manifolds.