Jacobi curvature operator v.s. Riemann curvature operator On Gravitation Page 286 Exercise 11.7, it mentioned a very interesting operator called "Jacobi curvature operator" 
$${\cal J}(u,v) n\equiv \frac{1}{2} [{\cal R}(n,u)v +{\cal R}(n,v)u ]$$
where it "contained the same information continent as Riemann", and 
in component form (Eq. 11.36) 
$$J^\mu_{\nu \alpha \beta} =\frac{1}{2} (R^\mu_{\alpha\nu\beta} +R^\mu_{\beta\nu\alpha})$$
It was also shown that Jacob could be directly used for geodesic deviation, since $${\cal J}(u,u) n ={\cal R}(n,u)u$$ a related post could be found here: Are the Jacobi equation and the geodesic deviation equation related?. 
However, why it seemed that Riemann was much popular than Jacobi, and that people seemed to use Jacobi much rarely? 
 A: Well, I would also see the Riemann tensor as a "primary" object and the Jacobi curvature tensor as the secondary one. This is readily seen in the definition of the Jacobi curvature tensor; i.e., it is the symmetrisation of the second and the fourth indices, on which Riemann tensor possesses no symmetry.
I think if you would see the Jacobi tensor as the primary definition then you would get into "problems" when you want to define it as a measure for the intrinsic curvature by parallel transporting a vector along a path. Therefore, you would have to introduce the Riemann curvature tensor again.
But, why they call it Jacobi curvature operator? I think the reason is that you could rewrite the Jacobi (see e.g. the post you added)
$$
 \nabla_u^2 n + {\cal R} (n ,u) u = 0
$$
in the symmetric form
$$
 \nabla_u^2 n +{\cal J}(u ,u) n = 0 \,.
$$
Then, one could write it as an operator acting on $n$:
$$
 [ \nabla_u^2  +{\cal J}(u ,u) ] n = 0 \,.
$$
It means the Jacobi equation will become a zero eigensolution problem.
You can see that this definition has other motivations than the Riemann curvature tensor. So I think the Riemann curvature tensor is indeed more fundamental.
Maybe there is a fundamental geometrical explanation for this. But, I'm not aware of that.
