The Levi Form

In previous chapters, concepts such as the tangential Cauchy–Riemann complex are introduced first for imbedded CR manifolds and then later for abstract CR manifolds. In this chapter, we take the opposite approach. First, we give the definition of the Levi form for the case of an abstract CR structure and then proceed to give more concrete representations of the Levi form in the case of an imbedded CR manifold. The Levi form for the case of a real hypersurface in ℂ ⁿ is discussed in some detail. In particular, the relationship between the Levi form and the first fundamental form of a hypersurface is presented.