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Is it possible to find $n>1$ such that $\mathbb{R}P^{2n+1}$ doesn't have smooth non vanishing vector field? I know it is not true for $S^{2n+1}$ and $\mathbb{R}P^{2n+1}$ is a sphere modolu antipod map, but I cannot see the answer thanks.

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The usual examples of smooth non-vanishing vector fields on odd-dimensional spheres seem to be invariant under the antipodal map so they descend to the projective space. –  Andreas Blass Jan 9 '13 at 19:09

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In the proof of Poincare-Hopf theorem, we see that a compact smooth manifold have a non-vanishing smooth tangent vector field if and only if it's Euler Character is $0$. And it's easy to prove that every odd-dimensional compact smooth manifold must have Euler Character $0$.

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