1 Introduction

Motivated by the well-known fact (see for instance [6]) that an almost complex structure \(\mathfrak{J}\) on a \(4\)-manifold \(X\) admits a description in terms of a rank \(2\) vector bundle \(\Lambda_\mathfrak{J}\subset \Lambda^2TX^*\), we introduce the notion of a splitting of an almost complex structure: A pair of line subbundles \(L_1,L_2\) of \(\Lambda^2TX^*\) is called a splitting of \(\mathfrak{J}\) if \(\Lambda_{\mathfrak{J}}=L_1\oplus L_2.\) A hypersurface \(M \subset X\) satisfying a nondegeneracy condition inherits a CR-structure from \(\mathfrak{J}\) and a path geometry from the splitting \((L_1,L_2).\) The purpose of this Note is to show that locally every real analytic path geometry is induced by an embedding into \(\mathbb{R}^4\simeq\mathbb{C}^2\) equipped with the splitting generated by the real and imaginary part of \(\mathrm{d}z^1\wedge \mathrm{d}z^2.\) This will be done using the Cartan-Kähler theorem. 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 \(\mathbb{C}^2.\) It follows with Nirenberg’s example of a smooth non-embeddable \(3\)-dimensional CR-manifold that the real analyticity in our main statement is necessary.

The notation and terminology for the Cartan-Kähler theorem and exterior differential systems are chosen to be consistent with [2, 7]. Moreover we adhere to the convention of summing over repeated indices.

Acknowledgment

Research for this article was carried out while the author was supported by Schweizerischer Nationalfonds SNF via the postdoctoral fellowship PBFRP2-133545 and by the Mathematical Sciences Research Institute, Berkeley.