非完整相空间的非结合几何与星R通量弦形变及（非）对称度量
Nonassociative geometry of nonholonomic phase spaces with star R-flux string deformations and (non) symmetric metrics
AI总结：
本文发展非完整相空间的非结合微分几何，利用星积与非完整标架构造R通量形变，并扩展至真空Einstein方程，涵盖对称与非对称度量。
摘要：
我们详细阐述了具有非完整（不可积）分布与标架、非线性与线性联络、对称与非对称度量，以及相应适配的拟Hopf代数结构的相空间的非结合微分几何。该方法基于为描述在恒定R通量背景中运动的闭弦而引入的非结合星积概念。我们考虑了广义Moyal-Weyl形变，其中对于星形变的非结合与非交换项，使用非完整标架（基底）代替局部偏导数。在此类修正的非结合与非完整时空及相关的复/实相空间中，几何与物理对象的系数既依赖于基础时空坐标，也依赖于常规的（余）纤维速度/动量变量，如同（非）交换Finsler-Lagrange-Hamilton几何中的情形。对于建模为Lorentz流形上（余）切丛总空间的非结合与（非）交换相空间，并配备星积与非完整标架，我们考虑了关联的非线性联络、N-联络结构，这些结构决定了常规的水平与（余）垂直（例如，4+4）维数分裂以及基本几何对象的N-适配分解。我们以抽象几何和N-适配系数形式定义并计算了挠率、曲率和Ricci张量。我们扩展了非完整几何的某些方法，以在N-适配线性联络以及对称与非对称度量结构的情形下构造真空Einstein方程的R通量形变。
英文摘要：
We elaborate on nonassociative differential geometry of phase spaces endowed with nonholonomic (non-integrable) distributions and frames, nonlinear and linear connections, symmetric and nonsymmetric metrics, and correspondingly adapted quasi-Hopf algebra structures. The approach is based on the concept of nonassociative star product introduced for describing closed strings moving in a constant R-flux background. Generalized Moyal-Weyl deformations are considered when, for nonassociative and noncommutative terms of star deformations, there are used nonholonomic frames (bases) instead of local partial derivatives. In such modified nonassociative and nonholonomic spacetimes and associated complex/ real phase spaces, the coefficients of geometric and physical objects depend both on base spacetime coordinates and conventional (co) fiber velocity/ momentum variables like in (non) commutative Finsler-Lagrange-Hamilton geometry. For nonassociative and (non) commutative phase spaces modelled as total spaces of (co) tangent bundles on Lorentz manifolds enabled with star products and nonholonomic frames, we consider associated nonlinear connection, N-connection, structures determining conventional horizontal and (co) vertical (for instance, 4+4) splitting of dimensions and N-adapted decompositions of fundamental geometric objects. There are defined and computed in abstract geometric and N-adapted coefficient forms the torsion, curvature and Ricci tensors. We extend certain methods of nonholonomic geometry in order to construct R-flux deformations of vacuum Einstein equations for the case of N-adapted linear connections and symmetric and nonsymmetric metric structures.
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