Explicit maximal totally real embeddings
Résumé
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 and Grauert. 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”. 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.
Domaines
Mathématiques [math]Origine | Fichiers produits par l'(les) auteur(s) |
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