Geodesics and the Gauß-Bonnet Theorem

In these notes we compute the geodesic curvature on a surface in isothermal coordinates and use it to prove the local Gauß-Bonnet Theorem. These remarks are a continuation of my notes [T] whose notation we continue to employ. 1. Isothermal Coordinates of a Surface. The computations are facilitated by using a special coordinate system in which the metric and the resulting formulas take a particularly simple form. Theorem [Isothermal Coordinates]. [Korn-Lichtenstein 1914, Lavrentiev, Morrey] Suppose M is a surface of class C (k ∈ N, 0 < α < 1) or C and P ∈ M . Then another coordinate patch X : Ω → M with P ∈ X(Ω) may be found (Ω is an open set of R2) such that X(u, u) ∈ C(Ω) or C(Ω), resp., such that the conformality relations g11 = X1 · X1 = g22 = X2 · X2 > 0 and g12 = X1 · X2 = 0 are satisfied for all (u, u) ∈ Ω. For a proof of this, see Jost [J]. Thus the metric g11 = g22 = φ(u , u) is given by a single function. The Gauss Curvature formula for orthogonal coordinates reduces to (1) K = − 1 2φ ∆logφ. where ∆ = ∂ 2 ∂(u) + ∂ ∂(u) is the usual Laplace operator. We shall have occasion to use the Christoffel symbols associated to this metric.