CR manifold

Summary

A CR manifold is a smooth real manifold with a distinguished involutive subbundle of its complexified tangent bundle, see the definition. This subbundle plays the role of the holomorphic tangent directions. The model is a real hypersurface in $\C^n$, such as the sphere $S^{2n-1}$, with the complex tangent directions that lie inside its tangent spaces. The letters CR stand for Cauchy–Riemann. A CR structure is what is needed to say which functions satisfy the tangential Cauchy–Riemann equations. CR geometry studies the boundaries of domains in several complex variables, and its main invariant is the Levi form.