显式极大全实嵌入
Explicit maximal totally real embeddings
AI总结：
该研究作为极大全实嵌入相关工作的续作，针对无挠实解析紧流形的典范复结构，给出了以曲率单项式对称化表示的纤维式泰勒展开及显式系数，拓展了论证适用范围并为可积方程的典范零化提供了佐证。
摘要：
本文是第一作者《论极大全实嵌入》一文的续作。对于配备了作用于其切丛实解析截面的协变导数算子的实解析紧流形，Guillemin-Stenzel、Lempert、Lempert-Szöke、Szöke 与 Bielawski 的前期研究已给出了典范极大全实嵌入的构造方法。该构造基于雅可比场（Jacobi fields）的运用，但其形式远非显式。因此，相应复结构的形式从一开始就是一个未解之谜。前述第一作者的论文中给出了这类复结构的一个相当简洁的递归表达式。我们的系列论文始终聚焦于无挠情形。在本文中，我们给出了该典范复结构的纤维式泰勒展开，其通过曲率单项式的对称化表示，且展开系数具有相当简洁显式的表达式。我们的核心论证可适用于更广泛的情形，这对研究所涉嵌入理论中的开放问题颇具助益。本文还为一般情形下部分可积方程的若干显著典范零化性质提供了佐证。
英文摘要：
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{ö}ke, Sz{ö}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.
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