The proofs I have seen that a vector field on the 2-sphere must have a zero rely on the general theorem that the index of any vector field on a manifold equals the manifold's Euler characteristic.(adsbygoogle = window.adsbygoogle || []).push({});

How about this for a proof that does not appeal to this general theorem?

The tangent circle bundle of the 2 sphere is homeomorphic to real projective 3 space.

A non-zero vector field would be a map from

[tex] S^2 -> RP^3 [/tex]

which has a left inverse, p, where p is just the bundle projection map.

Otherwise put,

p o v = identity on [tex] S^2 [/tex]

But the second real homology of projective 3 space is zero so p o v must equal zero on the fundamental cycle of the 2 sphere. This contradicts the equation p o v = identity.

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# Another proof that a vector field on the sphere must have a zero?

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