The equivalence problem for
CR structures can be viewed as a special case of the equivalence problem for
G-structure. This paper uses Cartan’s methods (in modernized form) to show that a
CR manifold of codimension 3 or greater with suitably generic Levi form
admits a canonical connection on a reduced structure bundle whose group is
isomorphic to the multiplicative group of complex numbers. As corollaries,
it follows that the CR manifold admits a canonical affine connection, and
consequently that the automorphisms of the CR manifold constitute a Lie
group.
The most difficult technical step is to construct a smooth moduli space for
generic vector-valued hermitian forms, which is tied to the CR manifold via the Levi
map. The techniques used to construct this space are drawn from the classical
invariant theory of complex projective hypersurfaces.
