# On maximal totally real embeddings

## TL;DR

This paper studies complex structures with totally real zero sections in tangent bundles, providing explicit integrability equations and fiberwise Taylor expansions under real-analytic assumptions, advancing understanding of their geometric properties.

## Contribution

It introduces explicit integrability equations and fiberwise Taylor expansions for complex structures with totally real zero sections, under real-analytic conditions.

## Key findings

- Explicit integrability equations derived
- Fiberwise Taylor expansions computed
- Enhanced understanding of complex structures in tangent bundles

## Abstract

9We 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 existence 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.