What is the geometric meaning of one Riemannian metric bigger than the other one on a smooth manifold? Gromov conjectured in 1985 and LLarull proved in 1998 that: If $g > g_0$ on the sphere, then there exists some point p on the sphere with $Sc(p) < Sc_0(p)$. Here $g, g_0$ are Riemannian metrics and $g_0$  is the standard metric on the sphere. $Sc$ is the scalar curvature. His proof is by contradiction and uses some non-vanishing index on the sphere.
Is the theorem also true for sectional curvature? Or is it the case that if one Riemannian metric is bigger than the other one on the sphere, then there must exist some point such that the scalar curvature at that point is smaller than the other one?
In general, is it true that the bigger the metric, the smaller the curvature (sectional, Ricci or scalar curvaure), at least in the pointwise setting?
 A: As it was stated by Phillip Andreae, Llarull's theorem implies that for some sectional direction sectional curvature is $< 1$ 
and for some unit vector Ricci curvature is smaller than $n-1$.
But if $g>g_0$ then $\mathop{\rm diam}(\mathbb{S}^n,g_1)>\pi$.
Therefore the cases of Ricci curvature (as well as sectional curvature) also follow from Myers's theorem.  
A: Lohkamp Scalar curvature and hammocks  proved that it it always possible to decrease both the metric and the scalar curvature simultaneously, as pointes out by Goette and Semmelmann in the paper Scalar curvature estimates for compact symmetric spaces. 
So it is not true that the bigger the metric, the smaller the scalar curvature in general.
The proof of Llarull's theorem only used the area-bigger (not the length of vector bigger) metric in the round sphere. Gomove call such metric are area extremal and asked which manifolds possess such an area extremal metric in Positive curvature, macroscopic dimension, spectral gaps and higher signatures.
As far as I know, the Gromov's question partially answered by Llarull, Min-Oo, Kramer, Goette and Semmelmann, and those examples are symmetric spaces.
Thus we can not say much about the geometric meaning of one Riemannian metric bigger than the other one on an arbitrarily smooth manifold, except  some symmetric spaces in above-mentioned.
