Explicit maximal totally real embeddings

This article deals with an explicit canonical construction of a maximal totally real embedding for real analytic manifolds equipped with a covariant derivative operator acting on the real analytic sections of its tangent bundle or of its complexified tangent bundle. The existence of maximal totally real embeddings for real analytic manifolds is known from previous celebrated works by Bruhat-Whitney [1] and Grauert [4]. Their construction is based on the use of analytic continuation of local frames and local coordinates that are far from being canonical or explicit. As a consequence, the form of the corresponding complex structure has been a mystery since the very beginning. A quite simple recursive expression for such complex structures has been provided in the first author's work “On maximal totally real embeddings” [12]. In our series of articles we focus on the case of torsion free connections. In the present article we give a fiberwise Taylor expansion of the canonical complex structure which is expressed in terms of symmetrization of curvature monomials and a rather simple and explicit expression of the coefficients of the expansion. We explain also a rather simple geometric characterization of such canonical complex structures. Our main result and argument can be useful for the study of open questions in the theory of the embeddings in consideration such as their moduli space.