Local embeddability of real analytic path geometries

TL;DR

This paper proves that every real analytic path geometry on a hypersurface in a 4-manifold can be locally realized through an embedding into complex 2-space, linking CR-structures and path geometries via Cartan-Kähler theory.

Contribution

It establishes a local embeddability result for real analytic path geometries using Cartan-Kähler theorem, connecting them to standard models in complex space.

Findings

Every real analytic path geometry is locally induced by an embedding into C^2.

Every 3D nondegenerate real analytic CR-structure can be locally embedded into C^2.

The approach uses Cartan-Kähler theory to achieve the embedding result.

Abstract

An almost complex structure J on a 4-manifold X may be described in terms of a rank 2 vector bundle E. A splitting of J consists of a pair of line bundles spanning E. A hypersurface M in X satisfying a nondegeneracy condition inherits a CR-structure from J and a path geometry from the splitting. Using the Cartan-K\"ahler theorem we show that locally every real analytic path geometry is induced by an embedding into C^2 equipped with the splitting generated by the real and imaginary part of the standard holomorphic volume form. 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 C^2.