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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $S^{2n} = \\set{x \\in \\R^{2n+1} : \\norm{x} = 1}$ be the [[sphere|T62369RFH6]] of even dimension $2n$.\nLet $v \\colon S^{2n} \\to \\R^{2n+1}$ be a continuous tangent [[vector field|0299VCW1TH]], that is, a continuous map with $v(x) \\cdot x = 0$ for all $x \\in S^{2n}$.\nThen $v(x) = 0$ for some $x \\in S^{2n}$.\n\n## Remarks\n\nSuppose $v$ has no zero, and put $w = v / \\norm{v}$.\nThen $H(x, t) = \\cos(\\pi t) \\, x + \\sin(\\pi t) \\, w(x)$ lies on the sphere and is a homotopy from the identity to the antipodal map.\nThe identity has [[degree|TMFG2P912R]] $1$ and the antipodal map of $S^{2n}$ has degree $(-1)^{2n+1} = -1$, which contradicts the homotopy invariance of degree.\n\nExample: model the wind on the Earth as a continuous tangent vector field on the sphere $S^2$.\nThen at any moment there is a point where the horizontal wind speed is zero.\n\nThe dimension must be even: on $S^{2n-1} \\subset \\R^{2n}$, the field $v(x_1, x_2, \\ldots, x_{2n-1}, x_{2n}) = (-x_2, x_1, \\ldots, -x_{2n}, x_{2n-1})$ is tangent and has no zero.\n\n## Explanations\n\n### L1 human\n\nYou cannot comb the hair on a ball flat without leaving a cowlick or a bald spot.\nIn mathematical terms, every continuous field of tangent arrows on an ordinary sphere has a point where the arrow is zero.\n\n### L2 human\n\nThe theorem follows from the degree of the antipodal map.\nIt is also the special case of the [[Poincaré–Hopf theorem|4XTPY82PP4]] for $S^{2n}$, since the Euler characteristic of $S^{2n}$ is $2 \\neq 0$.\nIt is the first case of the question how many linearly independent vector fields exist on a sphere, which Adams answered in the theorem on [[vector fields on spheres|3ES79NWRS7]].\nIt also implies that $S^{2n}$ carries no structure of a Lie group, since left translates of a nonzero tangent vector at the identity would give a nowhere vanishing vector field.\n",
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