Relationship Between Gradient of Distance Functions and Tangents to Geodesics

Proposition 1. Let C = (U,φ) be a coordinate chart on a open subset U of a Ddimensional manifold, Ω with coordinate variables u1,u2, · · · ,uD. Suppose U is Riemannian everywhere, equipped with a metric η (which is assume to be nonsingular everywhere i.e. η•• is positive definite), and the geodesic connecting any two points in U lies entirely in U. Let d :RD×RD→R be the distance function in U ⊆Ω in terms of the coordinate chart C (i.e. d(q,w), for q,w ∈ Img(φ) ⊆ RD, is the length of the shortest path connecting φ−1(q) and φ−1(w) in U ⊆Ω .). Suppose inside Img(φ), the distance d is induced by the Riemannian metric η , is smooth everywhere, and suppose there exists an unique geodesic of length d(q,w) connecting any two points q,w ∈ Img(φ). Then the following is true for every q,w ∈ Img(φ) ⊆ RD and every coordinate chart, C, defined on U (Figure 1(a)), [ ∂ ∂u d(q,u) ∣∣∣∣ u=w ]