 

Alan S. McRae
The GaussBonnet theorem for CayleyKlein geometries of dimension two


Published: 
July 18, 2006 
Keywords: 
CayleyKlein geometries, GaussBonnet theorem 
Subject: 
53C 


Abstract
We extend the classical GaussBonnet theorem for the Euclidean, elliptic, hyperbolic, and Lorentzian planes to the other three CayleyKlein geometries of dimension two, all three of which are absolutetime spacetimes, providing one proof for all nine geometries. Suppose that M is a polygon in any one of the nine geometries. Let Γ, the boundary of M, have length element ds, discontinuities θ_{i}, and signed geodesic curvature κ_{g}, where M and Γ are oriented according to Stokes' theorem. Let K denote the constant Gaussian curvature of the geometry with area form dA. Then
∫_{Γ} κ_{g} ds + ∑_{i} θ_{i} + ∫∫_{M} K dA = 2\pi
for the nonspacetimes and
∫_{Γ} κ_{g} ds + ∑_{i} θ_{i} + ∫∫_{M} K dA = 0
for the spacetimes, where we assume that Γ is timelike.


Author information
Department of Mathematics, Washington and Lee University, Lexington, VA 244500303
mcraea@wlu.edu

