 Consider the line subbundle of the complex cotangent bundle annihilating "V"

For each local section α ∈ Γ("H"0"M"), let

The form "h"α is a complex-valued hermitian form associated to α.

Generalizations of the Levi form exist when the manifold is not of hypersurface type, in which case the form no longer assumes values in a line bundle, but rather in a vector bundle.One may then speak, not of a Levi form, but of a collection of Levi forms for the structure.On abstract CR manifolds, of strongly pseudo-convex type, the Levi form gives rise to a pseudo-Hermitian metric.