Gauss–Bonnet theorem

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The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). It is named after Carl Friedrich Gauss who was aware of a version of the theorem but never published it, and Pierre Ossian Bonnet who published a special case in 1848.


Statement of the theorem

Suppose M is a compact two-dimensional Riemannian manifold with boundary \partial M. Let K be the Gaussian curvature of M, and let kg be the geodesic curvature of \partial M. Then

where dA is the element of area of the surface, and ds is the line element along the boundary of M. Here, χ(M) is the Euler characteristic of M.

If the boundary \partial M is piecewise smooth, then we interpret the integral \int_{\partial M}k_g\;ds as the sum of the corresponding integrals along the smooth portions of the boundary, plus the sum of the angles by which the smooth portions turn at the corners of the boundary.

Interpretation and significance

The theorem applies in particular to compact surfaces without boundary, in which case the integral \int_{\partial M}k_g\;ds can be omitted. It states that the total Gaussian curvature of such a closed surface is equal to 2π times the Euler characteristic of the surface. Note that for orientable compact surfaces without boundary, the Euler characteristic equals 2 − 2g, where g is the genus of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and g counts the number of handles.

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