• If
S is a compact surface (without boundary), then the inequality is an equality by the usual Gauss–Bonnet theorem for compact manifolds. • If
S has a boundary, then the Gauss–Bonnet theorem gives ::\iint_S K\, dA = 2\pi\chi(S) - \int_{\partial S}k_g\,ds :where k_g is the
geodesic curvature of the boundary, and its integral the
total curvature which is necessarily positive for a boundary curve, and the inequality is strict. (A similar result holds when the boundary of
S is piecewise smooth.) • If
S is the plane
R2, then the curvature of
S is zero, and
χ(
S) = 1, so the inequality is strict: 0 < 2. == Notes and references ==