There is hint: if M has isolated singular points, find a diffeomorphism to make these singular points in a any neighborhood which you want. How can we do next?
If a manifold M has zero Euler characteristic, there is a non-vanishing vector field on it
9
$\begingroup$
algebraic-topology
-
0@Javier That explains why zero Euler char is _necessary_ condition for existence of a non-vanishing vector field. But OP asks why it is _sufficient_ — and AFAICS the linked post doesn't help much. – 2011-06-25
-
0@Grigory M: Indeed, you are right!! Sorry, I read the problem backwards as I had just posted about index theorems giving as an amusing consequence the hairy ball theorem and thought it may be of conceptual help. Your obstruction solution is of much more help anyway. – 2011-06-25
-
1@Javier Actually... you were right, it's just [degree theory](http://math.stackexchange.com/questions/47451/consequences-of-degree-theory), essentially. – 2011-06-26