The Gaussian curvature value $G$ is the product of the two principle curvatures, which are the largest, $\kappa_1$, and the smallest (most downwards-curving), $\kappa_2$, normal curvatures through a point): $$G=\kappa_1\kappa_2$$

For a mountain top as well as a valley, we have a positive $G>0$, because both principle curvatures are either upwards or downwards. The same is the case for a sphere (rightmost figure below). A saddle point, on the other hand, will have negative $G<0$ (leftmost figure). And a flat surface as well as a surface which is 'flat' in one dimension, such as a cylinder (middle figure), will have $G=0$.

enter image description hereImage from Wikipedia

My question is regarding a specific claim from the article on Gaussian curvatures on Wikipedia ("Relation to geometries" headline, middle sentence):

When a surface has a constant zero Gaussian curvature, then it is a developable surface and the geometry of the surface is Euclidean geometry.

When a surface has a constant positive Gaussian curvature, then it is a sphere and the geometry of the surface is spherical geometry.

When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface and the geometry of the surface is hyperbolic geometry.

These three sentences seemingly cover all types of surfaces. But the middle sentence (I have emphasized it) tells that for a positive $G>0$, the curvature is a sphere. How can that be true? Don't many mountain top surfaces have a positive $G>0$ all the way down from the top to forever? Such as the function $f=x^2+y^2$ (graph below)?

enter image description here

from WolframAlpha

  • $\begingroup$ constant positive curvature $\endgroup$ – user10354138 May 13 '19 at 16:00
  • $\begingroup$ @user10354138 Oh, right. I forgot for a moment what constant means... Thank you for pointing me to the answer. $\endgroup$ – Steeven May 13 '19 at 16:02
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    $\begingroup$ Wikipedia is sloppy here. The correct result is that if $(S,g)$ is a simply-connected compact Riemannian surface of constant positive curvature then it is isometric to a round sphere of some radius. $\endgroup$ – Moishe Kohan May 13 '19 at 16:29

A simple comment by @user10354138 gave the answer. I simply didn't care enough for the meaning of constant. It is now clear for me, and I will here answer and close the question.

As pointed out in the comments, the keyword here is constant positive Gaussian curvature. While a mountain top surface such as $f=x^2+y^2$ has a positive Gaussian curvature $G$ at all points, the value of $G$ changes and becomes less positive for points farther from the stationary point (the peak), because the curvatures widen and become "less sharp" as the surface broadens out away from its peak.

A constant $G$ requires the same type of curvature at any point. The same shape as seen from every point. That is clearly only the case for a sphere. Wikipedia is right.

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