Using the theory of exterior differential systems, we study the existence of germs of almost holomorphic disk in a real analytic hypersurface locally defined in a complex manifold equipped with J a real analytic almost complex structure. The integrable case in \({\mathbb {C}}^n\) with J the multiplication by i has been intensively studied by several authors (D’Angelo, Ann Math (2) 115(3):615—637, 1982; D’Angelo, Several complex variables and the geometry of real hypersurfaces. Studies in advanced mathematics. CRC Press, Boca Raton, 1993, p xiv+272; Diederich and Fornaess, Ann Math (2) 107(2):371–384, 1978) for example. The non integrable case is drastically different essentially due to the following fact: in generic case, there is no J-invariant objects of dimension bigger than one. This simple observation leads to the non existence of some equivalents of Segree varieties or ideals of holomorphic functions which play a fundamental role in the complex case. Nevertheless in the almost complex case, we adopt the exterior differential system point of view of E. Cartan developed and clarified in Bryant et al. (Exterior differential systems. Springer, Berlin, 1991).
