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If a flat surface disc is replaced by a smooth disc agreeing with the original surface outside the same boundary, their boundary geodesic curvatures agree. Gauss-Bonnet theorem therefore forces the replacement’s total Gaussian curvature to be zero. If its curvature is everywhere nonnegative, continuity forces it to vanish everywhere. Thus a positive-curvature bulge cannot be glued this way without some negative curvature or a loss of smoothness.

A surface homeomorphic to a cylinder with strictly negative Gaussian curvature has at most one closed geodesic image. Geodesic disks, monogons and bigons are excluded by Gauss-Bonnet theorem; distinct essential simple geodesics would bound an annulus whose curvature integral must vanish. Repeated traversals count as the same image.