I'm reading Gromov's article "Four lectures on scalar curvature". On the page 6, he claims that the map $g\mapsto Sc(g)$ is linear in the second derivatives of $g$. I don't know the exact meaning of such statement.
Here is my comprehension：In local coordinate, we have $$Sc(g)=\dfrac{1}{2}g^{ij}g^{kl}\left(\dfrac{\partial^2 g_{il}}{\partial x^k\partial x^j}+\dfrac{\partial^2 g_{kj}}{\partial x^i\partial x^l}-\dfrac{\partial^2 g_{kl}}{\partial x^i\partial x^j}-\dfrac{\partial^2 g_{ij}}{\partial x^k\partial x^l}\right)+g^{ij}\Gamma_{ij}^m\Gamma_{km}^k-g^{ij}\Gamma_{kj}^m\Gamma_{im}^k$$ In this expression, I regard $Sc$ as a function $Sc(g,g',g'')$ and the second derivatives of $g$ are only contained in the first term. Then I get $$Sc(ag,ag',ag'')=aSc(g,g',g'')$$It seems some kind of linearity.
Besides, at the same position, Gromov claims that the map $Sc$ is determined uniquely by two peoperties up to a scalar multiplication. I don't know how to prove it.
The article can be found at arXiv:1908.10612v6.
