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Let's say we have an arbitrary tangent vector field on $S^n$. Can just say that the tangent vector field is continuous?

Edit: I saw this in Hatcher's book that "$S^n$ has a continuous field of tangent vectors iff $n$ is odd" and wondered if "continuous" term is guaranteed without mentioning it or not.

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Why? Of course not: Pick any point and a nonzero vector in the tangent space to that point and set the vector field to be zero everywhere else. Your question is the same as asking: let's say we have a function on $[0,1]$... – t.b. Apr 4 '12 at 16:05
Where are you seeing this term? Most authors working with manifolds probably only use the term "vector field" to mean a smooth one. I'd say this question depends on context. – Matt Apr 4 '12 at 17:03
up vote 1 down vote accepted

I think it's often taken to be part of the definition of "vector field" that it's continuous. The fact that you can't comb the hair on a billiard ball, i.e. there is no non-vanishing field of tangent vectors on $S^2$, would obviously not be true if you didn't take continuity to be part of the definition of "vector field".

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