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arXiv · 2608.14686

Introduction to Differential Geometry in the Three-Dimensional Field of Complex Vectors

Branko Sarić
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Abstract

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.

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Branko Sarić. 2026-08-05. Introduction to Differential Geometry in the Three-Dimensional Field of Complex Vectors. https://arxiv.org/abs/2608.14686

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