Non-Riemannian Geometry (COLLOQUIUM PUBLICATIONS (AMER MATHEMATICAL SOC))

The use of the differential geometry of a Riemannian space in the mathematical formulation of physical theories led to important developments in the geometry of such spaces. The concept of parallelism of vectors, as introduced by Levi-Civita, gave rise to a theory of the affine properties of a Riemannian space. Covariant differentiation, as developed by Christoffel and Ricci, is a fundamental process in this theory. Various writers, notably Eddington, Einstein and Weyl, in their efforts to formulate a combined theory of gravitation and electromagnetism, proposed a simultaneous generalization of this process and of the definition of parallelism. This generalization consisted in using general functions of the coordinates in the formulas of covariant differentiation in place of the Christoffel symbols formed with respect to the fundamental tensor of a Riemannian space. This has been the line of approach adopted also by Cartan, Schouten and others. When such a set of functions is assigned to a space it is said to be affinely connected. From the affine point of view the geodesics of a Riemannian space are the straight lines, in the sense that the tangents to a geodesic are parallel with respect to the curve. In any affinely connected space there are straight lines, which we call the paths. A path is uniquely determined by a point and a direction or by two points within a sufficiently restricted region. Conversely, a system of curves possessing this property may be taken as the straight lines of a space and an affine connection deduced therefrom. This method of departure was adopted by Veblen and Eisenhart in their papers dealing with the geometry of paths, the equations of the paths being a generalization of those of geodesics by the process described in the first paragraph. In presenting the development of these ideas Eisenhart begins with a definition of covariant differentiation which involves functions $L^i_{jk}$ of the coordinates, the law connecting the corresponding functions in any two coordinate systems being fundamental. Upon this foundation a general tensor calculus was built and a theory of parallelism.