I have been confused about this for some time. For example, the curvature tensor in general for the Levi-Civita connection of $g$ in the local frame $\{\partial_i\}$ induced by the coordinates $\{x^i\}$ is
$R_{ijk}^l = \partial_i \Gamma_{jk}^l – \partial_j \Gamma_{ik}^l + \Gamma_{jk}^m \Gamma_{im}^l – \Gamma_{ik}^m\Gamma_{jm}^l$
and if you take the partials of the Christoffel symbols and grind through you can ultimately get
$R_{ijk}^l = \frac{1}{2}g^{ls}(\partial_i \partial_k g_{js} – \partial_i \partial_s g_{jk} -\partial_j \partial_k g_{is} + \partial_j\partial_s g_{ik}) + g^{lm}g_{st}(\Gamma_{jm}^s\Gamma_{ik}^t – \Gamma_{im}^s\Gamma_{jk}^t)$
which is fine. But in every book I read everyone just says "at a point $p$ in normal coordinates we have…"
$R_{ijk}^l = \frac{1}{2}g^{ls}(\partial_i \partial_k g_{js} – \partial_i \partial_s g_{jk} -\partial_j \partial_k g_{is} + \partial_j\partial_s g_{ik})$
which of course follows right away from what I already have, but what is the point of this second formula at all? Are they trying to tell me that if I change coordinates back to the coordinate frame in the latter formula that I'm supposed to get the former? I've been trying to do this with no success, am I failing because I'm making some computational mistake or because my logic is wrong altogether?
Since $R$ is a tensor its value at a point $p$ does not depend on the coordinates:
Does this mean that if $\tilde\partial_j$ is the frame induced by the coordinates $\{y^j\}$ then $g(R(\partial_i, \partial_j) \partial_k, \partial_l))=\tilde{g}(\tilde{R}(\tilde\partial_i, \tilde\partial_j)\tilde\partial_k,\tilde\partial_l)$ as real numbers at every point $p$ in the overlap of the two charts? Does this mean that if for some reason I wanted to integrate the component $R_{123}^4$ over some compact $M$ I could just integrate its formula in normal coordinates?
If the questions seem vague its because I really am confused by this. Thanks for whatever clarification you can give.
Best Answer
As object is said to be coordinate independent when it can be defined without direct reference to coordinate charts. For instance, the Riemannian curvature $R$ can be viewed as a multilinear map between tangent spaces $$ R|_{p}:T_pM\times T_pM\times T_pM\to T_pM \\ (X_p,Y_p,Z_p)\mapsto R(X_p,Y_p)Z_p $$ And similarly for other tensors. This does not mean that the components $R^{a}{}_{bcd}$ will be independent of the choice of coordintes (though they will be related by a simple transformation rule). A component such as $R^4{}_{123}$ has no meaning unless we fix a coordinate chart. Index expressions remain useful because they hold in any coordinate chart, even though the values of the individual components may change.
Generally speaking, coordinates are most often used as an intermediate step in computing coordinate-independent relationships between coordinate-independent objects. In this context, one can choose an arbitrary coordinate chart, or one which is convenient for the task at hand.
For instance, if we want to prove the first Bianchi identity $$ R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0 $$ we can work one point at a time by choosing an arbitrary point $p$ and showing it holds at $p$. By choosing normal coordinates centered at $p$, we can do the computation in a setting where the components of $R$ are particularly simple, since half of the terms vanish.