coord = {t, r, θ, ϕ};
n  = Length[coord];

gMatrix = 
        DiagonalMatrix[{
(1 -  
                Rs/r), 
- 1/
                c^2  1/(1 -  Rs/ r), 
 -1/
                c^2 r^2, 
 -1/
                c^2 r^2 (Sin[θ])^2
}];
gMatrix  // MatrixForm

g[i_, 
        j_] :=  gMatrix[[i]][[j]] 
gInv[i_, 
        j_] :=  Inverse[gMatrix][[i]][[j]]

GammaDown[i_,  j_, k_] :=  
      1/2 ( 
D[g[i, k], 
              coord[[j]]] 
 +  
            D[g[i, j], coord[[k]]] 
 - 
            D[g[j, k], coord[[i]]] )

GammaUp[i_, j_, k_] := 
      Sum[ 
gInv[i, s] GammaDown[s, j, 
            k],  
{s, 1, n}]

NonZeroChristoffelSymbols  = {};

    
\(For[i = 1, \ 
      i ≤ 
        n, 
For[j = 1, \ 
          j ≤ 
            n, 
For[k = j, \ 
              k ≤ n, 
GammaTmp\  = 
                Simplify[
                  GammaUp[i, j, 
                    k]]; 

 (*\ 
                In\ the\ next\ line\  <> \ 
                    is\ the\ JoinString\ operator\ *) 
GammaStatement\  = {ToString["Γ^"]
 <> 
                    ToString[
                      coord[\([i]\)]]
 <> \ "_"
 <> ToString[coord[\([j]\)]]
 <> 
                    ToString[coord[\([k]\)]]
 <> 
                    ToString[\ " ="], 
GammaTmp}; 

 (*\ 
                Using\ the\ Simplify\ command\ we\ can\ check\ if\ an\ expression\ is\ zero\ *) \ 
addCond = 
                Simplify[
                  GammaTmp == 
                    0]; 

 (*\ 
                If\ the\ expression\ is\ non - zero, \ 
                add\ it\ to\ list\ of\ non - 
                  zero\ symbols\ *) 
\ \ If[
                addCond == 
                  True, , , 
\(NonZeroChristoffelSymbols = 
                    Append[NonZeroChristoffelSymbols, 
                      GammaStatement];\)
]; 
\(k++\)
]
\(j++\)
]
\(i++\)
]\)
NonZeroChristoffelSymbols  // MatrixForm

R[b_, a_, d_, 
        c_] := 
D[GammaUp[c, b, d], 
          coord[[a]]] - 
D[GammaUp[c, a, d], 
          coord[[b]]] + 
Sum[
          GammaUp[e, b, d] GammaUp[c, e, a], {e, 1, 
            n}] - 
Sum[
          GammaUp[e, a, d] GammaUp[c, e, b], {e, 1, n}]

Ricci[a_,  b_] :=  Sum[ R[a, i, b, i], {i, 1, n}]

Table[
Simplify[
          Ricci[a, b]], 
{a, 1, n}, {b, 1, 
          n}]//MatrixForm