Local embeddability of real analytic path geometries (Q417656)

It is known that an almost complex structure \(\mathfrak J\) on a 4-manifold \(X\) can be described using a rank-2 vector bundle \(\Lambda_{\mathfrak J} \subset \Lambda^2TX^*\). A pair of line subbundles \(L_1\) and \(L_2\) is called a splitting if \(\Lambda_{\mathfrak J} = L_1 \oplus L_2\). Then every hypersurface \(M \subset X\) satisfying a nondegeneracy condition inherits a path geometry from the splitting. NEWLINENEWLINENEWLINENEWLINE The main result of the paper is to show that given a real-analytic path geometry on a manifold \(M\), it is locally realizable by a real-analytic embedding into \({\mathbb{C}}^2 \cong {\mathbb{R}}^4\) with the splitting generated by the real and imaginary part of \(dz^1 \wedge dz^2\). That real-analyticity is necessary follows from the smooth nonembeddable example of Nirenberg. A corollary of the main result is the well known fact that every 3 dimensional nongenerate real-analytic CR-structure is induced locally by an embedding into \({\mathbb{C}}^2\).