Reference of what metric can be placed on manifold?

Question

I just read some conclusion that $T^2$ can't be placed metric with positive curvature at all points. I don't know why is so . And what book introduce about this ? I mean about what metric can be placed after given some manifold.

Answer 1

This follows from Gauss-Bonnet Theorem: If $f$ is the Gaussian curvature of a compact surface $S$ without boundary, then $$\int_S f=2\pi\chi(S)$$ where $\chi(S)$ is the Euler characteristics of $S$. In particular, if $S$ is $T^2$ the torus, we have $\chi(S)=\chi(T^2)=0$. Therefore, it is impossible for $f>0$ everywhere.

BTW, for higher dimensional torus $T^n$, it is proved that its Yamabe constant is zero (due to Schoen and Yau?), which shows that it cannot be equipped with metric with positive scalar curvature at every point.

Answer 2

By the Gauss–Bonnet theorem, the Euler characteristic of $T^2$ is given by $$\chi(T^2) = \frac{1}{2\pi} \int_{T^2} K dA$$ where $K$ is the curvature and $dA$ is the element area of $T^2$. If $K$ were everywhere positive, then this would be a positive number, for the same reason that the integral of a positive function is positive; but $\chi(T^2) = \chi(S^1) \cdot \chi(S^1)$ is zero. Thus $K$ cannot be everywhere positive (and in fact if it's positive somewhere then it has to be negative at some other position).