| 1. | The Gaussian curvature coincides with the sectional curvature of the surface.
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| 2. | At the same time, a plane has zero Gaussian curvature.
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| 3. | A flat umbilic is an umbilic with zero Gaussian curvature.
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| 4. | This gives another way of establishing the intrinsic nature of Gaussian curvature.
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| 5. | Formally, Gaussian curvature only depends on the Riemannian metric of the surface.
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| 6. | Like fractals, it has no defined Gaussian curvature.
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| 7. | These are doubly ruled surfaces of negative Gaussian curvature.
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| 8. | The sphere and the plane have different Gaussian curvatures, so this is impossible.
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| 9. | We see that projective transformations don't mix Gaussian curvatures of different sign.
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| 10. | The uniformization theorem for universal covering space of surfaces with constant negative Gaussian curvature.
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