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- W2186806621 abstract "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 Kd A =2 π for the nonspacetimes and � Γ κ g ds + � i θi + ��" @default.
- W2186806621 created "2016-06-24" @default.
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- W2186806621 date "2006-01-01" @default.
- W2186806621 modified "2023-09-27" @default.
- W2186806621 title "The Gauss-Bonnet theorem for Cayley-Klein geometries of dimension two" @default.
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