A Constructive Demonstration of the Uniqueness of the Christoffel Symbol

On the other hand, whether the Christoffel symbol is the only connection which can be constructed from a symmetric second–rank tensor gμν remains an open question. In this note we exhibit a constructive demonstration of the uniqueness of the Christoffel symbol. Let us start by reviewing some simple results of tensor calculus. Let M be an n–dimensional differentiable manifold. Several geometric objects can be introduced over conveniently defined fibered bundles based on M. In order to classify them we adopt a taxonomic approach, cf. (Visconti, 1992): a tensor is an object which transforms like a tensor, etc. The previous definition makes reference to the way in which an object transforms under changes of local coordinates. Let x, μ = 1, 2, · · · , n, and y, α = 1, 2, · · · , n, be local coordinates on M. Both sets are related by y = y(x), and the differential form of this relation is