New York Journal of Mathematics
Volume 12 (2006) 143-155


Alan S. McRae

The Gauss-Bonnet theorem for Cayley-Klein geometries of dimension two

Published: July 18, 2006
Keywords: Cayley-Klein geometries, Gauss-Bonnet theorem
Subject: 53C

We extend the classical Gauss-Bonnet theorem for the Euclidean, elliptic, hyperbolic, and Lorentzian planes to the other three Cayley-Klein geometries of dimension two, all three of which are absolute-time 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 θi, and signed geodesic curvature κg, 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 κg ds + ∑i θi + ∫∫M K dA = 2\pi
for the nonspacetimes and
Γ κg ds + ∑i θi + ∫∫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 24450-0303