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Wikipedia says:

"For surfaces with genus $0$ with isolated umbilics, e.g. an ellipsoid, the index of the principal direction vector field must be $2$ by the Poincaré–Hopf theorem. Generic genus $0$ surfaces have at least four umbilics of index $\frac{1}{2}$. An ellipsoid of revolution has two non-generic umbilics each of which has index $1$."

What is the difference between an ellipsoid and an ellipsoid of revolution? Why does an ellipsoid have at least four, but an ellipsoid has only two?

Ng Chung Tak
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Hao Wu
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1 Answers1

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An ellipsoid has three perpendicular axes of arbitrary lengths.

An ellipsoid of revolution is a special case where two of these three axes are the same length, as it has circular cross-sections; it is also called a spheroid. The points at the ends of the third axis will be umbilical

In the even more special case of a sphere where all three axes are equal, all the points are umbilical

Henry
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  • Then wikipedia on one point seems wrong to say there must be at least four umbilical points on an ellipsoid. It should be just four umbilical points, right? If wikipedia on this point is right, how to see there are five or more umbilical points on an ellipsoid? – Hao Wu May 05 '17 at 16:08