Is there an intuitive geometric proof to this?
2026-03-28 04:23:22.1774671802
Any tangent vector field on S^2 has a singular point
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If a singular point of a tangent vector field $X$ means that $X(x)=0$, your assertion is not true. There exists parallelizable spheres, $S^1,S^3,S^7$ which are the only parallelizable spheres after results of Bott, Kervaire, Milnor.