DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§1. Holonomy and the Gauss-Bonnet Theorem 83To see this, we reason as follows. Given a boundary vertex v, denote by a superscript (v) therelevant angle or number for which the vertex v is involved. When v is a fixed smooth boundaryvertex, we haveπ = ∑ι (v)λ= ∑(π − ɛ (v)λ)=π(E(v) ib+1)− ∑ɛ (v)λ .v∈∆ λ v∈∆ λ v∈∆ λOn the other hand, when v is a fixed corner of ∂M with exterior angle ɛ k ,wehaveɛ k = π − ∑ι (v)λ= π − ∑(π − ɛ (v)λ)= ∑ɛ (v)λ− πE(v) ib .v∈∆ λ v∈∆ λ v∈∆ λThus,∑boundaryverticesɛ λj =∑v cornerɛ (v)λ + ∑v smoothɛ (v)λAdding equations (∗) and (∗∗) yields∑ɛ λj =λ,j∑interiorverticesɛ λj +( l∑= ɛ k +k=1∑boundaryvertices∑v corner)πE (v)ib+ ∑πE (v)ib=v smoothɛ λj =2π(E i + E ib − V i )+l∑ɛ k .k=1l∑ɛ k + πE ib .At long last, therefore, our reckoning concludes:∫ ∫∫l∑κ g ds + KdA+ ɛ k =2π ( )F − (E i + E ib )+V i =2π(V − E + F )=2πχ(M,T),∂MMk=1because we can deduce from the fact that the boundary curve ∂M is closed that V b = E b .We now derive some interesting conclusions:Corollary 1.8. The Euler characteristic χ(M,T) does not depend on the triangulation T ofM.Proof. The left-hand-side of the equality in Theorem 1.7 has nothing whatsoever to do withthe triangulation. □It is therefore legitimate to denote the Euler characteristic by χ(M), with no reference to thetriangulation. It is proved in a course in algebraic topology that the Euler characteristic is a“topological invariant”; i.e., if we deform the surface M in a bijective, continuous manner (so as toobtain a homeomorphic surface), the Euler characteristic does not change. We therefore deduce:Corollary 1.9. The quantity∫∂M∫∫κ g ds + KdA+Ml∑ɛ kk=1is a topological invariant, i.e., does not change as we deform the surface M.In particular, in the event that ∂M = ∅, sothe surface M is closed, wehaveCorollary 1.10. When M is a closed, oriented surface without boundary, we have∫∫KdA =2πχ(M).Mk=1□