TITLE: How to understand $R^{\rho}_{\sigma\mu\nu}$ for constructing geometries?
QUESTION [5 upvotes]: I'm trying to understand the curvature tensor $R^{\rho}_{\sigma\mu\nu}$ by playing with it in certain contexts. I already have an understanding, to some degree, of how it measures the change in a vector as it moves along a path in curved space, and how it's related to parallel transport. The issue is that there are just too many components for me to understand succinctly which parts correspond to what and why, in detail.
I tried constructing a differential equation with the intention of having the curvature of a manifold embedded into Euclidean space decrease as a function of some Euclidean distance from a point source. I began by writing
$$u^{\alpha}\partial_{\alpha}R^{\rho}_{\sigma\mu\nu}=$$
where $u$ is a vector pointing towards the source, and am at a loss for what the right hand side should be. Fundamentally, why does this curvature tensor need as many components as is does to describe something which is a relatively simple concept? Also any help with the differential equation above would be nice. Again, I'm trying to build a solid concept of this thing, so what is the best way to understand it?

REPLY [4 votes]: First, the last two indices are skew-symmetric and therefore the curvature tensor can be viewed as a $2$-form that depends on two indices,
$$
\Omega^i{}_j = R^i{}_{jkl}\,dx^k\wedge dx^l.
$$
Therefore, for given $i$ and $j$, $\Omega^i{}_j$ is something that can be integrated along a surface. Second, if written with respect to an orthonormal frame $(e_1, \dots, e_n)$, the tensor is also skew-symmetric with respect to the indices $i$ and $j$. Here, that should be interpreted as say that, for given $k$ and $l$,
$$R^i{}_{jkl}\,e_i\otimes \omega^j$$ (where $(\omega^1, \dots, \omega^n)$ is the dual basis to $(e_1, \dots, e_n)$) is $o(n)$-valued, where $o(n)$ is the Lie algebra of the orthogonal group $O(n)$. In other words, for given $k$ and $l$, the curvature tensor is an infinitesimal rotation.
In particular, it measures the infinitesimal rotation of the given orthonormal frame relative to parallel transation. And, if you integrate the curvature $2$-form along a surface with boundary in $M$, you get the actual difference between the orthonormal frame at a point on the boundary and its parallel translate around the boundary. This is known as the Ambrose-Singer theorem.
A rather terse explanation for this (for the more general case of a connection on a vector bundle) can be found here: https://deaneyang.github.io/blog/blog/math/differential-geometry/riemannian-geometry/2020/08/07/Holonomy.html