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.

中文翻译：

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具有星型R磁通字符串变形和（非）对称度量的非完整相空间的非缔合几何

我们详细阐述了具有非完整（不可积分）分布和框架，非线性和线性连接，对称和非对称度量以及相应适应的拟霍夫代数结构的相空间的非缔合微分几何。该方法基于引入的非缔合星积的概念，用于描述在恒定R磁通背景下移动的闭合弦。对于星形变形的非缔合和非交换项，使用非完整框架（基）代替局部偏导数时，将考虑广义Moyal-Weyl变形。在这样修改的非缔合和非完整时空以及关联的复/实相空间中，几何和物理对象的系数取决于基本时空坐标和常规（共）纤维速度/动量变量，例如（非）可交换Finsler-Lagrange-Hamilton几何形状。对于建模为具有星积和非完整框架的Lorentz流形上（co）切线束的总空间建模的非缔合和（非）交换相空间，我们考虑关联的非线性连接，N连接，确定常规水平和（co）垂直的结构（例如4 + 4）的尺寸分割和基本几何对象的N自适应分解。有抽象的几何定义和计算，N适应系数形成了扭转，曲率和Ricci张量。