I have a question about the topic in the headline.
We have defined a oriented manifold as a manifold with a atlas, s.t. the determinant of the differential of chart changes is >0, i.e. det forall x,y charts.
I think it is true that if we have a non orientable manifold, then the der above has to be 0 for some charts.
To put it another way, if the manifold is not or., then it can't appear that der(...)<0.
My questions are:
1)Is my conjecture correct?
2)Do you know a argument, why there has to be a atlas, s.t. det(..)>0 if we have a atlas s.t. det(..)<0?
I try to put a "-" to each chart. putthen the "-" cancel out, since we have the composition of thwo charts.