Gauss-Bonnet theorem

(Carl Friedrich Gauss and Pierre Ossian Bonnet) Given a two-dimensional compactPlanetmathPlanetmath Riemannian manifoldMathworldPlanetmath M with boundary, Gaussian curvature of points G and geodesic curvature of points gx on the boundary M, it is the case that


where χ(M) is the Euler characteristicMathworldPlanetmath of the manifoldMathworldPlanetmath, dA denotes the measure with respect to area, and ds denotes the measure with respect to arclength on the boundary. This theorem expresses a topological invariantPlanetmathPlanetmath in terms of geometrical information.

Title Gauss-Bonnet theorem
Canonical name GaussBonnetTheorem
Date of creation 2013-03-22 16:36:37
Last modified on 2013-03-22 16:36:37
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Theorem
Classification msc 53A05