Complex 2-Dimensional Internal Space in General Relativity

A calculus of vectors in 2‐dimensional symplectic spaces is developed from the concept of existence of local basis systems. The similarities, as well as the differences, of this calculus with the tetrad formulation of 4‐dimensional curved spaces are discussed. The affinity and curvature of the symplectic space are derived and its relationships with the affinity and curvature of the usual spinor formalism are given. A system of hybrid geometrical objects displaying a tensor and a spinor index take over the role of the usual Hermitian matrices ${\sigma }_{\mu }^{\mathrm{KṀ}}\mathrm{(x).}$