It is clear that the closed connected manifold admit non-vanishing vector field if and only if Euler characteristic of the manifold is zero. Let manifold M is non-compact or compact with boundary. Is it true M admits a non-vanishing vector field?
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1See https://math.stackexchange.com/questions/586862/non-vanishing-vector-fields-on-non-compact-manifolds . – Travis Willse Nov 20 '18 at 14:26
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1See also https://mathoverflow.net/questions/52023/is-there-a-poincare-hopf-index-theorem-for-non-compact-manifolds . – Travis Willse Nov 20 '18 at 14:27