There are many questions about what metric can be placed on a given manifold . For example , place a metric with non-negative curvature.
As I know , the Gauss-Bonnet theory is useful in this question. But in some question , the Euler characteristic is not zero, then ,it is not useful for whether can be placed metric with non-negative or non-positive curvature. In fact ,I don't know the general method about this kind of question .
So, I think whether we can do it by studying the maps from given manifold to suitable manifold ? Are there any theory about this ?
This just be a soft question ,and my English is poor. So ,if there are any problem , tell me .