In Riemannian Geometry (or Pseudo-Riemannian too), all of the operations performed are in general position dependent. For example, the metric tensor $g_{\mu\nu}(x)$ is dependent on position $x^i$ for $i=0,\cdots,N$.
Since the raising and lowering of components in a rank-n tensor is based on the metric tensor then the act of raising and lowering of components seems to be position dependent as well (my assumption, please confirm true/false).
And, I am also assuming that even performing a contraction is position dependent.
Therefore, I am assuming that when you are actually working on a problem, computing actual numbers for a solution that all of these operations that we normally display merely by index manipulation are position dependent and therefore computation-heavy operations.
However, if this assumption is correct then I am confused about the general case of deriving the Ricci Tensor from the Riemannian Curvature tensor via contraction operation. Sure, given a well formed coordinate system (Spherical?) such operations are maintained in functional form with variables $r, \theta, and\, \phi$ but is this always the case? Or, is this contraction producing the Ricci tensor performed for each individual position in the domain of computation?
Please confirm or correct my assumptions. Alternatively, point me to some resource that describes how one goes about computing with tensors in the general sense (where an obviously simple Euclidean space is not assumed).