On maximal totally real embeddings

Abstract

Abstract We consider complex structures with totally real zero section of the tangent bundle. We assume that the complex structure tensor is real-analytic along the fibers of the tangent bundle. This assumption is quite natural in view of a well-known result by Bruhat and Whitney. We provide explicit integrability equations for such complex structures in terms of the fiberwise Taylor expansion. In a particular geometric case considered in the literature, we explicit much further the fiberwise Taylor expansion of the complex structure as well as the integrability equations.