Suppose M is a compact twodimensional orientable Riemannian manifold with boundary ∂M. Denote by K the Gaussian curvature[?] at points of M, and by k_{g} the geodesic curvature[?] at points of ∂M. Then
The theorem applies in particular if the manifold does not have a boundary, in which case the integral ∫_{∂M} k_{g} ds can be omitted.
If one bends and deforms the manifold M, its Euler characteristic will not change, while the curvatures at given points will. The theorem requires, somewhat surprisingly, that the total integral of all curvatures will remain the same.
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