Riemannian Manifolds

Abstract

The metric tensor having been introduced, the notions of orthogonality and local parallelism may then be introduced by applying the classical Euclidean laws to infinitesimal triangles. For this purpose, the laws of similar triangles must be adopted ab initio as postulates, and the ‘postulate of parallels’ must be excluded, a procedure that is reasonable as far as physics is concerned both because the laws of similar triangles correspond immediately to the intuition of experience and because experience is always limited to finite regions. By defining the right angle as the angle of intersection of two lines that makes all four intersection angles equal, and by guaranteeing its uniqueness through further axiomatic refinements on the comparison of angles by means of the notions ‘greater than’ and ‘less than’ as well as ‘equality’, one may then derive the Pythagorean theorem in the well-known manner.