Talk:Umbilical point

What if both curvatures are zero?

"The sphere is the only surface where every point is umbilic."

What about the plane? —Simetrical (talk • contribs) 19:22, 6 April 2009 (UTC)[reply]

Well spotted, indeed every Developable surface has everywhere zero curvature so are also everywhere umbilic. --Salix (talk): 20:04, 6 April 2009 (UTC)[reply]

(Apparent?) contradiction

The article says close to the beginning: "For surfaces with genus 0, e.g. an ellipsoid, there must be at least four umbilics,[citation needed] a consequence of the Poincaré–Hopf theorem. An ellipsoid of revolution has only two umbilics." Now, if I'm not mistaken, an ellipsoid of rotation IS an example of a surface of genus 0. However, "at least four" does not include "two". Where is the problem? I strongly suspect it's in the article which hence needs fixing.WikiPidi (talk) 08:21, 19 November 2019 (UTC)[reply]

With an ellipsoid of revolution, the poles actually have more symmetry than a normal umbilic, you could call these higher umbilics. If you were to slightly deform the surface you would find the points broke up into multiple umbilics.

In terms of the Poincaré–Hopf theorem the total index of a vector field on a genus 0 surface must 2. If we use the vector field from one set of principal directions, normal umbilics either have index +1/2 (Lemon/Monstar) or -1/2 (Star). So for typical genus 0 surfaces there must be at least four Lemon/Monstar umbilics. Now if we consider the umbilic on an ellipsoid of revolution. Here the vector field will be symmetric with all principle directions pointing towards the umbilic, having an index of +1.

A refined version of the statement which currently is

For surfaces with genus 0, e.g. an ellipsoid, there must be at least four umbilics,[citation needed] a consequence of the Poincaré–Hopf theorem. An ellipsoid of revolution has only two umbilics.

might be

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

--Salix alba (talk): 11:53, 19 November 2019 (UTC)[reply]