<<xAct`xCoba`

DefManifold[S2,2,{a,b,c,d,e,f,g,h}]
DefMetric[1,metric[-a,-b],CD,{";","∇"},PrintAs->"g"]
DefChart[RS,S2,{0,1},{u[],v[]}]
MatrixForm[metricarray=4/(1+u[]^2+v[]^2)^2 {{1,0},{0,1}}]
MetricInBasis[metric,-RS,metricarray]
MetricCompute[metric,RS,All,CVSimplify->Simplify]

DefTensor[x[a],S2]

$PrePrint=ScreenDollarIndices;

ScalarsOfChart@RS

x[a]==Inner[Times,ScalarsOfChart@RS,ComponentArray@Basis[a,-{b,RS}],Plus]

ToBasis[RS]/@%

ComponentArray@%

%/.Equal->ComponentValue

Separate basis has to be used twice in order to get Christoffel symbols replaced.

(CD[-a][x[b]])//SeparateBasis[RS]//SeparateBasis[RS]//TraceBasisDummy//ToValues