hairy ball thm. and projective space 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 modulo antipod map, but I cannot see the answer 
thanks.

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 $0$. 
