An Intrinsic Proof of the Gauss-bonnet Theorem

LetM be a 2-dimensional compact oriented Riemannian manifold. We start by choosing an open subset U of M on which we can define an orthonormal frame (E1, E2). Recall that this means that E1, E2 are vector fields on U such that 〈Ei, Ej〉 = δij (the Kronecker delta), where 〈·, ·〉 denotes the Riemannian metric on M . We sometimes call (E1, E2) a moving frame or repère mobile in French, and this method of studying manifolds, the method of moving frames (la méthode de repère mobile). The method was invented by G. Darboux and used extensively by E. Cartan. Let (ω1, ω2) be the coframe associated with (E1, E2). Recall that this means that ω1, ω2 are differential 1-forms on U such that ωi(Ej) = δij . 1.1. Remark. Observe that on U , ω1 ∧ω2 = dA, the Riemannian volume form of M , since ω1 ∧ω2 is a 2-form (recall that dimM = 2) and (ω1 ∧ ω2)(E1, E2) = ω1(E1)ω2(E2)− ω1(E2)ω2(E1) = 1. So even though the individual forms ω1, ω2 depend on the choice of frame, ω1 ∧ ω2 does not. 1.2. Lemma (Levi-Civita). There exist unique 1-forms ω12 and ω21, with ω21 = −ω12 such that dω1 = ω12 ∧ ω2 and dω2 = ω21 ∧ ω1. (1)