It is well known that vanishing of Riemann curvature tensor is equivalent to the existance of coordinate system where Christoffel Symbols are zero. That is to say:
$R^a_{bcd}=0 \iff \exists\text{ Coord Sys }(x): \Gamma^i_{jk}(x)=0$
Is it a way to state a similar about a-th component of this tensor? That is: $R^i_{bcd}=0 \iff \exists\text{ Coord Sys }(x): \Gamma^i_{jk}(x)=0$,
for any i ?
For example:
$R^1_{bcd}=0 \iff \exists\text{ Coord Sys }(x): \Gamma^1_{jk}(x)=0$,
but the rest are unrestricted.
Since we have:
$R^i_{jkl}=\frac{\partial \Gamma^i_{jl}}{\partial x^k}-\frac{\partial \Gamma^i_{jk}}{\partial x^l}+\Gamma^s_{jl}\Gamma^i_{ks}-\Gamma^s_{jk}\Gamma^i_{ls}$,
the implication that $R^i_{bcd}=0$ when $\Gamma^i_{jk}=0$ is trivial. How can I prove (or disprove) an implication in the oposite way.
I will be gratefull for any suggestions!
PS. What is the proper mathematical symbol to express: "a coordinate system with coordinates $(x^1...x^n)$"? I don't like the way I've done it here.