The penetration of vectors in differential geometry (Q1203022)

The paper discusses how conceptual difficulties and prejudices about the nature of mathematics made the introduction of vectors in differential geometry difficult for about a century. Differential geometry as a general study of curves and surfaces in the three dimensional space with respect to their curvature started with the beginnings of differential calculus. The beginnings of vector geometry or of a calculus to operate directly with physico-geometrical objects and not with their numerical coordinates started in the middle of the nineteenth century. Soon after, several mathematicians like Burali Forti and Henri Fehr advocated the use of vector geometry in the teaching of differential geometry. However they had no success because of the influence of renowned mathematicians like Bianchi and Darboux who considered the vector approach ``abstract'' and ``nonintuitive''. In his much praised book on differential geometry Bianchi however dedicated two chapters on \(n\)-dimensional Riemannian geometry with the concept of tensor. In the treatment of three dimensional differential geometry however there was no mention of vector or tensor. This was the situation till 1910, mainly because of the intuitive appeal of three dimensional geometry both in mathematics and physics. The situation changed drastically when Einstein showed that many phenomena of modern physics could best be explained by a four dimensional Riemannian geometry, showing that an ordered set of numbers has a particular physical and geometrical significance. The idea of ``invariant under a transformation'' which became extraordinarily significant for geometry and physics showed the conceptual similarity of vectors and tensors. Modern authors of differential geometry at once introduce vectors to explain the concepts of differential geometry, thus preferring the abstract approach as opposed to the intuitive approach of Darboux and Bianchi. In the development of mathematics the abstract approach has prevailed because it can unify several disciplines in mathematics. Whether, however, as regards learning and understanding is concerned, the intuitive approach of Darboux and Bianchi was not a better ``first step'' than the vector approach remains an interesting and an open question.