Reduction of Five-Dimensional Uniformly Levi Degenerate CR Structures to Absolute Parallelisms

An almost CR-structure on a smooth manifoldM is a subbundle H(M) ⊂ T (M) of the tangent bundle of even rank endowed with operators of complex structure Jp : Hp(M) → Hp(M), J p = −id, that smoothly depend on p. A manifold equipped with an almost CR-structure is called an almost CR-manifold. The subspaces Hp(M) are called the complex tangent spaces to M , and their complex dimension, denoted by CRdimM , is the CR-dimension of M . The complementary dimension CRcodimM := dimM − 2CRdimM is called the CR-codimension of M . Further, a smooth map f : M → M̃ between two almost CR-manifolds is a CR-map if for every p ∈ M the differential df(p) of f at p maps Hp(M) into Hf(p)(M̃) and is complexlinear on Hp(M). If for two almost CR-manifolds M , M̃ of equal CR-dimensions there exists a diffeomorphism f from M onto M̃ that is also a CR map, then the manifolds are said to be CR-equivalent and f is called a CR-isomorphism. We are interested, in particular, in the equivalence problem for almost CR-manifolds. This problem can be viewed as a special case of the equivalence problem for G-structures. Let G ⊂ GL(d,R) be a Lie subgroup. A G-structure on a d-dimensional manifold M is a subbundle S of the frame bundle F (M) over M that is a principal G-bundle. Two G-structures S, S̃ on manifolds M , M̃ , respectively, are called equivalent if there is a G-structure isomorphism between S, S̃, i.e. a diffeomorphism f from M onto M̃ such that the induced mapping f∗ : F (M) → F (M̃) maps S onto S̃. The almost CR-structure of a manifoldM of CR-dimension n and CR-codimension k is a G-structure with G being the group of all nondegenerate linear transformations of Cn⊕Rk that preserve the first component and are complex-linear on it. The notion of equivalence of such G-structures is then exactly that of almost CR-structures. For convenience, when speaking about G-structures below, we replace the frame bundle F (M) by the coframe bundle. Important examples of G-structures are {e}-structures, where {e} is the one-element group. They are called absolute parallelisms, and on a d-dimensional manifold M any such structure is given by an R-valued 1-form that for every p ∈ M defines an isomorphism between Tp(M) and R. The (local) equivalence problem for absolute parallelisms is well-understood (see e.g. p. 344 in [Ste]), and therefore one may approach the equivalence problem for general G-structures by attempting to reduce them to absolute parallelisms. Let C be a class of manifolds equipped with G-structures. The G-structures in C are said to reduce to absolute