AI 中文总结
本文基于复向量二维、三维场的交换几何代数，构建了嵌入复向量三维场的紧致埃尔米特流形微分几何向量框架，推导相关张量关系，为复流形微分几何提供了统一代数几何基础。
AI 中文摘要
本文基于已为复向量二维和三维场开发的交换几何代数，构建了嵌入复向量三维环境场中的紧致埃尔米特流形的微分几何向量框架。该基础代数框架由复向量的交换几何乘积构建，在二维和三维场景中均产生统一的交换代数及对应的向量微分与积分恒等式。这些代数结果为独立于经典标量坐标方法开发的微分几何公式提供了内在基础。在该框架内，紧致埃尔米特流形完全以向量形式描述，推导了支配复流形内在与外在几何的基本张量关系。所得框架建立了统一的代数与几何场景，可直接从复向量的交换代数出发发展复流形的微分几何。
英文摘要
Building on a commutative geometric algebra developed for 2D and 3D fields of complex vectors, this paper develops a vector framework for the differential geometry of compact Hermitian manifolds embedded in an ambient three-dimensional field of complex vectors. The underlying algebraic framework is constructed from the commutative geometric product of complex vectors, yielding a unified commutative algebra in both the 2D and 3D settings together with the corresponding vector differential and integral identities. These algebraic results provide the intrinsic foundation for a differential geometric formulation that is developed independently of the classical scalar coordinate approach. Within this framework, compact Hermitian manifolds are described entirely in vector form. Fundamental tensor relations governing the intrinsic and extrinsic geometry of complex manifolds are derived in vector form. The resulting framework establishes a unified algebraic and geometric setting in which the differential geometry of complex manifolds is developed directly from the commutative algebra of complex vectors.
Comments16 pages
