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I have an exercise which says that :extend the Hopf-Poincare theorem for non-orientable manifold with the indication using the double covering. I have got stuck for long time, so I don't know if somebody can help me. Any answer is greatly appreciated.

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1) Pull the vector field back to the double cover 2) apply the orientable Poincaré-Hopf theorem 3) divide by two. Since the index is local in nature, this won't change the index of a zero (and each original zero corresponds to two zeroes on the double cover). The Euler characteristic is multiplicative with respect to $n$-sheeted coverings. –  commenter Oct 17 '12 at 16:16
Thanks! I get a lot! –  mapping Oct 18 '12 at 8:32

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