Q:

Indices of the Riemann tensor

Recently I have been using a video guide/explanation type thing on YouTube to gain a better understanding of the Riemann tensor and it's links to the Ricci tensor: https://youtu.be/sBDGPIuJ-HA . I understand the formulation of the tensor using a parallelogram he uses however and just in general how the tensor is formed. However I'm unsure as to what indices mean, how the dimensions are managed and how contraction works- specifically. 
Is it like the stress energy tensor where certain information is kept in certain indices, such that information concerning volume deviation is contained in certain indices, meaning those indices survive the contraction and remain in the Ricci tensor?
I guess my real question is where do the indices come from and what do they represent?

A:

You should familiarize yourself with a geometric interpretation of the Riemann-tensor:
The Riemann-tensor expresses the path-dependence of parallel transport.
Now, best way to ascertain path-dependence is to parallel transport a vector around a closed loop. Because the curvature tensor depends on only one point (wherever it is evaulated), the curvature tensor cannot provide a finite description of parallel transport around a closed loop, only infinitesimal.
Let us define the index structure of the Riemann tensor now as $$ R_{ab\ \ d}^{\ \ \ \ c}=\partial_a\Gamma^c_{bd}-\partial_b\Gamma^c_{ad}+\Gamma^c_{ae}\Gamma^e_{bd}-\Gamma^c_{be}\Gamma^e_{ad}$$ or as $$ R_{ab\ \ d}^{\ \ \ \ c}X^d=(\nabla_a\nabla_b-\nabla_b\nabla_a)X^c. $$
This index structure is precisely what is necessary to express the concept of infinitesimal parallel transport around a loop, it is because we need
1) a loop,
2) a linear transformation that describes the change in the vector.

On a manifold, vectors do not describe finite displacements, unlike in euclidean geometry, but they do describe infinitesimal displacements, so we can take two vectors, $v^a$ and $w^a$ to span an infinitesimal parallelogram. But as you know (cross products, determinants), generalized, signed area measures are antisymmetric, so $v^aw^b-w^av^b$ is what describes the "infinitesimal parallelogram" spanned by $v$ and $w$. $$  $$
For this reason, the first two indices of the Riemann-tensor are antisymmetric, because they "accept" a parallelogram: $$ R_{ab\ \ d}^{\ \ \ \ c}v^aw^b. $$ Now we get an object that has two indices remaining: $$ \mathfrak{O}^c_{\ d}=R_{ab\ \ d}^{\ \ \ \ c}v^aw^b. $$

Now that we have a loop, we need a linear transformation. A lin. transformation is provided by a tensor with one up and one down index, so $\mathfrak{O}^c_{\ d}$ is our transformation. The expression $$ \mathfrak{O}^c_{\ d}X^d $$ describes the transformation the vector $X$ undergoes, as it is parallel transported along the closed loop spanned by $v$ and $w$. However the Levi-Civita connection (eg. the covariant derivative described by Christoffel-symbols) preserves the lengths of vectors during parallel transport, so the vector $X$ can only change by rotation. $$  $$ This gives us the reason why the Riemann-tensor is antisymmetric in the $c,d$ indices as well (provided they are simultaneously raised or lowered). If you have a one-parameter family of rotations $O_\epsilon$ such that $O_\epsilon=1$ (the rotation at $\epsilon=0$ is the identity), then to first order in $\epsilon$ we have $$ O_\epsilon=1+\epsilon \mathfrak{O}, $$ where $\mathfrak{O}$ is antisymmetric. If instead of euclidean rotations, we consider rotations of a metric of signature (-+++) (these are essentially Lorentz-transforms), then the "antisymmetry" property applies only if this matrix has its indics raised or lowered.

In conclusion, $$ R_{ab\ \ d}^{\ \ \ \ c}v^aw^bX^d $$ describes the infinitesimal change the $X$ vector picks up, when it is parallel transported along the infinitesimal parallelogram spanned by $v$ and $w$. We need two antisymmetric indices to give a parallelogram ($a$ and $b$) and one index to act on a the vector to be transported ($d$). That's 3 indices, but the end result of the parallel transport is a vector itself, so that's the fourth index ($c$).

