Historical Remarks on Gauss–Bonnet

Abstract

Publisher Summary This chapter discusses historical remarks on Gauss Bonnet. If M is a two-dimensional oriented Riemannian manifold and D is a compact domain on M bounded by a sectionally smooth curve C, the Gauss–Bonnet formula states that ∑ (π – α) + ∫ c k g ds + X D KdA = 2πX(D), where X( D ) is the Euler characteristic of D and the members at the left-hand side are respectively the exterior angles at the corners, the integral of the geodesic curvature, and the integral of the Gaussian curvature. These are respectively the point, line, and surface curvatures, so that the formula expresses the total curvature in terms of a topological invariant. The most important special case of the formula is the theorem on the angle sum of a rectilinear triangle in the euclidean plane.