The Ricci Calculus

Abstract

We now begin our presentation of the Ricci calculus, or the calculus of congruences of curves, which will form one of the essential ingredients of the leg calculus. Our exposition closely follows the classical one given by Ricci which is recounted in the books of Levi-Cività (1925), Eisenhart (1926), and Weatherburn (1938), except for our use of language and some organizational changes. These changes are intended to ease the inclusion of the material into the leg calculus in Chapter IV. Essentially, relative to terminology, the usage in the classical literature is not uniform, and a system of orthonormal unit tangent vectors, which for us is a vectorial n-leg {λ_a}, was called a pyramid (an n-dimensional generalization of a trihedron in 3-dimensions) by Levi-Cività, and an orthogonal ennuple by Eisenhart and Weatherburn. We regard such terminology as being obsolete and inferior to that of an n-leg which explicitly exhibits the dimensionality of the system. Likewise, these authors, following Ricci, set out the theory in an n-dimensional Riemannian space V _n , while for our purposes we need only the case n = 3. Hence, we will consider only the case of a curved Riemannian V₃ and ultimately specialize it to a flat Euclidean E₃.