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What is the geometrical "meaning" of gaussian curvature?

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The most geometrical answer I know of is the Gauss–Bonnet theorem. One particular case of it tells us that the sum of the angles of a geodesic triangle on a surface is equal to $\displaystyle \pi + \int_\Delta K \, dA$, where the integral is over the interior of the triangle and $K$ is the Gaussian curvature.

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