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A line segment has zero normal curvature in its tangent direction, impossible when Gaussian curvature is positive because both principal curvatures have the same nonzero sign. Two disjoint closed geodesics would bound an annulus. Applying Gauss-Bonnet theorem, its geodesic boundary contributes zero and $\chi =0$, but $\int KdA>0$, a contradiction.

A triangular prism is the orthogonal Cartesian product of a triangle with a line segment.