# Number of zeros of (tangent) vector field on sphere

## Main Question or Discussion Point

Is it possible to have a tangent vector field on the unit 2-sphere x^2+y^2+z^2 =1 in
3D which vanishes at exactly one point? By the Poincare-Hopf index theorem
the index of such vector field at the point where it vanishes must be 2. Is that possible? If yes, can one write an explicit formula for such vector field.

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Hurkyl
Staff Emeritus
Gold Member
Yes. Observe that your question is equivalent to "Does there exist a tangent vector field on R^2 that vanishes at infinity?" Does that give you any ideas?

Yes, thank you for the hint. I see it now. I was able to compute the following explicit formula for the vector field:

(x^2/(z-1) + 1, x*y/(z-1), x)

for any (x,y,z) with x^2+y^2+z^2=1 except (0,0,1).

mathwonk
Homework Helper
stick one finger in your eye and a thumb in the other and bring thumb and finger together.

that gives a vector field with a single zero in the middle of your nose, of index 2.

All that gave me was a headache...

mathwonk