= Smooth nonnegative-curvature replacement of a flat disc is flat
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.
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