Local embeddability of real analytic path geometries

An almost complex structure JJ on a 4-manifold X   may be described in terms of a rank 2 vector bundle ΛJ⊂Λ2TX⁎ΛJ⊂Λ2TX⁎. We call a pair of line subbundles L1L1, L2L2 of Λ2TX⁎Λ2TX⁎ a splitting of JJ if ΛJ=L1⊕L2ΛJ=L1⊕L2. A hypersurface M⊂XM⊂X satisfying a nondegeneracy condition inherits a CR-structure from JJ and a path geometry from the splitting (L1,L2)(L1,L2). Using the Cartan–Kähler theorem we show that locally every real analytic path geometry is induced by an embedding into C2C2 equipped with the splitting generated by the real and imaginary part of dz1∧dz2dz1∧dz2. As a corollary we obtain the well-known fact that every 3-dimensional nondegenerate real analytic CR-structure is locally induced by an embedding into C2C2.