Local embeddability of real analytic path geometries

Details Local embeddability of real analytic path geometries Local embeddability of real analytic path geometries

2012-02-06

arxiv.org/abs/1202.1250v2

Differential Geom. Appl. 30 (2012), no. 2, 222-226

Mathematics

Differential Geometry

8 pages. Version 2 corrects some typos

Scientific paper

10.1016/j.difgeo.2012.03.002

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.

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