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Because the sign is constant.

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What if the curvatures vanish?

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As a very rough intuitive guide, suppose that you have a picture of anHow do you tell from a picture of a surface what the signs of the mean and gauss curvature are?

The Gaussian curvature is the product of the two principal curvatures, so it is positive if these have the same sign, negative if they have opposite signs, and zero if either of them is zero. From that, it should be obvious that the Gaussian curvature is negative (at each point) for the object on the left in the picture, zero for the cylinder in the centre of the picture, and positive for the ball on the right.

The mean curvature is the arithmetic mean of the two principal curvatures. So it is positive if they are both positive. If they have opposite signs then you have to judge which of them is larger in absolute value.

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