Let $(M,g)$ be a SR-manifold and for $p \in M$, let $\{e_1,\ldots,e_n\}$ be a pseudo-orthonormal basis for $T_pM$. Now, in my notes, it says that, if $v \in T_pM$ is such that $g(v,v) \neq 0$ and $$\{e_2,\ldots,e_n\}$$ is an orthonormal basis of $v^{\perp}$ then $$\operatorname{Ric}(v,v) = g(v,v) \sum_{i = 2}^{n} K(v,e_i)$$ where $K(-,-)$ is the sectional curvature of non-degenerate two-planes at $p \in M$. Then it says that from this, we can interpret $\operatorname{Ric}(v,v)$ as the ”mean” of all sectional curvatures of two-planes containing $v$.
Now, for the scalar curvature $\operatorname{scal} = \operatorname{tr}_g(\operatorname{Ric}) \in C^{\infty}(M)$ we have $$\operatorname{scal}(p) = \sum_{i,j = 1}^{n} \epsilon_i \epsilon_j R(e_j,e_i,e_i,e_j) = \sum_{\substack{i,j = 1, \\ i \neq j}}^{n} \epsilon_i \epsilon_j K(e_i,e_j)(\underbrace{g(e_i,e_i)}_{=\epsilon_i}\underbrace{g(e_j,e_j)}_{=\epsilon_j}-\underbrace{g(e_i,e_j)}_{= 0}) = \sum_{i \neq j} K(e_i,e_j)$$ so that $\operatorname{scal}(p)$ is the ”mean” over all $i \neq j$ sectional curvatures of non-degenerate two-planes.
My question is this: In what sense are these two descriptions ”means”? My mind is naturally drawn to the arithmetic mean, but there are of course other means, and I am not sure there is a mathematical definition of mean independent of the specific mean one is referring to, but since we don’t divide by something, it can’t be the arithmetic mean, I presume. Any thoughts on in which sense these are means? And for the scalar curvature, I presume they must mean all non-degenerate two-planes spanned by the basis vectors? Because could there not potentially be other two-planes?
Best Answer
It's the arithmetic mean, up to a constant multiple. If $v$ is a unit vector, then $\operatorname{Ric}(v,v)/(n-1)$ is the average of the sectional curvatures of basis $2$-planes containing $v$. Similarly, if you divide the scalar curvature by $n(n-1)$, you get the average of all the sectional curvatures of basis $2$-planes.
You can turn this into a more invariant kind of average that doesn't depend on a choice of basis: If you think of the set of all $2$-planes containing a unit vector $v\in T_pM$ as being parametrized by the set of unit vectors orthogonal to $v$ (an $(n-2)$-sphere), then you can show that $\operatorname{Ric}(v,v)/(n-1)$ is equal to the integral of sectional curvatures over that sphere divided by the volume of the sphere. Similarly, $\operatorname{scal}/(n(n-1))$ is equal to the integral of sectional curvatures over the Grassmannian of oriented $2$-planes in $T_pM$, divided by the volume of that Grassmannian.