PlanetMath: Gauss-Bonnet theorem

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Gauss-Bonnet theorem

(Theorem)

(Carl Friedrich Gauss and Pierre Ossian Bonnet) Given a two-dimensional compact Riemannian manifold with boundary, Gaussian curvature of points and geodesic curvature of points on the boundary $\partial M$ , it is the case that

$\displaystyle \int_M G \, dA + \int_{\partial M}g_x ds = 2\pi\chi(M),$

where $\chi(M)$ is the Euler characteristic of the manifold,

denotes the measure with respect to area, and

denotes the measure with respect to arclength on the boundary. This theorem expresses a topological invariant in terms of geometrical information.

"Gauss-Bonnet theorem" is owned by rspuzio. [ full author list (2) | owner history (2) ]

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53A05 (Differential geometry :: Classical differential geometry :: Surfaces in Euclidean space)

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