|
In differential geometry, Cohn-Vossen's inequality, named after Stephan Cohn-Vossen, relates the integral of Gaussian curvature of a non-compact surface to the Euler characteristic. It is akin to the Gauss–Bonnet theorem for a compact surface. A divergent path within a Riemannian manifold is a smooth curve in the manifold that is not contained within any compact subset of the manifold. A complete manifold is one in which every divergent path has infinite length with respect to the Riemannian metric on the manifold. Cohn-Vossen's inequality states that in every complete Riemannian 2-manifold ''S'' with finite total curvature and finite Euler characteristic, we have〔Robert Osserman, ''A Survey of Minimal Surfaces'', Courier Dover Publications, 2002, page 86.〕 : where ''K'' is the Gaussian curvature, ''dA'' is the element of area, and ''χ'' is the Euler characteristic. ==Examples== * 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 :: :where 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cohn-Vossen's inequality」の詳細全文を読む スポンサード リンク
|