Let U R^2 be diffeomorphic to a closed disk with smooth boundary given by a simp

Question

Let U R^2 be diffeomorphic to a closed disk with smooth boundary given by a simply closed regular curve gamma: R rightarrow U (gamma is periodic with period L). Let g be any Riemannian metric on U with curvature function K_g. Since gamma is regular we may assume that gamma is arclength parameterized g(gamma'. gamma') = 1. Let N: I rightarrowR^2 be of unit length and normal to gamma w.r.t. to the Riemannian metric g so that the ON-frame {gamma, N} along gamma has the standard orientation. Convince yourself that nabla_gamma^gamma is normal to gamma (w.r.t. g). The geodesic curvature k_g of gamma then is defined by nabla_gamma^gamma = k_g N Prove the local Gauss-Bonnet theorem integral_U K_g d vol_g + integral_0^L k_g dt = 2 pi

Explanation / Answer

Geodesic curvature for a unit speed curve on a surface, the length of the surface tangential component of acceleration is the geodesic curvature. Curves with are called geodesics. For a curve parameterized.

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