A Bianchi’s Relation Between Commutative Hypercomplex Numbers and Riemannian Geometry: an Updated Reading

Abstract.

In a paper published in 1916, the mathematician L. Bianchi has shown that the commutative hypercomplex numbers can be related with Riemannian spaces with null curvature (Euclidean spaces).

After summarizing this paper we take into account the developments from the time of publication, in particular the formalization of the theory of functions of a hypercomplex variable that allows us to complement and simplify the solution of the problem.

Afterwards we look for a physical interpretation of the results: beginning with the two-dimensional number systems, we give to Bianchi’s transformation the physical meaning of a field generated by a point source. In particular the field associated with hyperbolic numbers can be related with the equivalence principle, one of the Einstein’s starting point for the formulation of general relativity.