@Grisha, any closed manifold times with 2-sphere admits metrics with positive scalar curvature.

@BenMcKay, if the metric comes from Riemannian metric, then it is right. If not, (for example, CAT(0) space), how to do it?

@Paul, thanks for your information. On the sphere, g ≥ g_st means the inner product of g bounded below by the stand one in every point for every two tanget vectors. It looks like need the curvature condition to give the lower bound π of the conjugate raduis.

@Anton Petrunin, I am sorry that I don't make it clear. It is for all 2-planes in my mind as Mizar and Phillip Andrease said. What I really want to understand is described in the title. Geometry is decided by the distance if we are "back to Eulid". Therefore, the negative answer of "the bigger the metric, the smaller the curvature" surprised me.