Explicit maximal totally real embeddings

TL;DR

This paper develops an explicit recursive formula for the canonical complex structure on maximal totally real embeddings of real analytic manifolds, extending previous implicit constructions and providing new insights into their integrability properties.

Contribution

It introduces a fiberwise Taylor expansion of the complex structure expressed via curvature monomials, making the structure more explicit and applicable to broader settings.

Findings

Explicit recursive formula for complex structures

Taylor expansion expressed in curvature monomials

Evidence of vanishing integrability equations

Abstract

This article is the continuation of the first named author work "On maximal totally real embeddings". For real analytic compact manifolds equipped with a covariant derivative operator acting on the real analytic sections of its tangent bundle, a construction of canonical maximal totally real embeddings is known from previous works by Guillemin-Stenzel, Lempert, Lempert-Sz{\"o}ke, Sz{\"o}ke and Bielawski. The construction is based on the use of Jacobi fields, which are far from being 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 above cited first named author work. In our series of papers we always focus on the torsion free case. In the present paper we provide 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. Our main argument applies to far more generals settings that can be useful for the study of open questions in the theory of the embeddings in consideration. In this paper we provide also evidence for some remarkable canonical vanishing of some of the integrability equations in general settings.