Hopf Conjecture states that:
If even-dimensional manifold $M$ admit a metric of positive (non-negative) curvature then its Euler characteristic is positive (non-negative).
My question is about non-negative case. What is the meaning of non-negative Euler characteristic? I think "non-negative Euler characteristic" must be "zero Euler characteristic" or does there exist manifolds of non-negative curvature of positive and zero Euler characteristic?
Nonnegative means $\ge 0$. Thus, one part of the conjecture reads: $K_M\ge 0$ $\Rightarrow$ $\chi(M)\ge 0$. The other part reads $K_M> 0$ $\Rightarrow$ $\chi(M)> 0$. You cannot expect $K_M\ge 0$ $\Rightarrow$ $\chi(M)= 0$ to hold. For instance, think about $S^2\times S^2$ or even $M=S^4$. Also, $K_M\ge 0$ does not imply $\chi(M)>0$. Standard examples would be $M=T^4$ and $M=S^3\times S^1$.