How can I prove that there exists no non vanishing vector field on $S^2$. I know there is a general theorem, called "Hairy Ball Theorem". But since I am a beginner, I have not yet studied all the things which are needed for the proof of Hairy Ball Theorem. I have studied basics of manifold, smooth maps, tangent spaces, homotopy theory. Is there any proof based on these topics. Thank you very much.
2026-04-08 09:06:51.1775639211
there exists no non vanishing vector field on $S^2$
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