Introduction to Differential Geometry in the 3D Field of Complex Vectors

Based on the isomorphic algebraic structures of the 2D Euclidean field of complex vectors V_{ℂ} and the field of complex numbers ℂ, in terms of identical geometric products of the elements of both fields, this paper, in the first three sections, brings the algebraic structure of a 3D field of complex vectors, as well as the corresponding fundamental integral identities in those vector fields. Additionally, in the fourth section, fundamental vector relations are presented, which relate to a compact Hermitian manifold embedded in an ambient 3D field of complex vectors.