The Levi form and the notion of pseudo-convexity have played an important role in the study of several complex variables and complex manifolds, [1] and . Several complex variables uses the notion of pseudo-convexity in an analytic manner, namely, for a priori estimates, whereas in complex manifolds the Levi form is used as a substitute for the second fundamental form. This paper studies the geometric significance of the Levi form and of the idea of pseudo-convexity and attempts to place these ideas in the general framework of classical differential geometry.
The first section of the paper introduces the Levi form in an invariant manner and relates it to the analytic definition. Section 2 derives a Steiner type formula for the volume of a collar neighborhood around a real hypersurface as a polynomial in the length of the collar with integral invariant coefficients. At this point the elementary symmetric functions of the Levi curvatures make the largest contribution to the integral invariants. Section 3 focuses on the interplay between curvature and Betti numbers and discusses several theorems comparing pseudo-convex hypersurfaces with spheres. For example, under suitable conditions, a finite submanifold of a Kaehler manifold is shown to be diffeomorphic to a standard disc.