$M$ is the 4D spacetime.
$\Sigma$ is the 3D spatial hypersurface.

Riemann Curvature Tensor:
${R^a}_{bcd} = {\Gamma^a}_{bd,c} - {\Gamma^a}_{bc,d} + {\Gamma^a}_{ec} {\Gamma^e}_{bd} - {\Gamma^a}_{ed} {\Gamma^e}_{bc} - {\Gamma^a}_{be} {c_{cd}}^e$

Commutation coefficients:
${c_{ab}}^c = {\Gamma^c}_{ba} - {\Gamma^c}_{ab}$

Substitute and expand:
${R^a}_{bcd} = {\Gamma^a}_{bd,c} - {\Gamma^a}_{bc,d} + {\Gamma^a}_{ec} {\Gamma^e}_{bd} - {\Gamma^a}_{ed} {\Gamma^e}_{bc} - {\Gamma^a}_{be} {\Gamma^e}_{dc} + {\Gamma^a}_{be} {\Gamma^e}_{cd}$

Split into Space+Time:
${R^a}_{btt} = 0$
${R^a}_{bti} = {\Gamma^a}_{bi,t} - {\Gamma^a}_{bt,i} + {\Gamma^a}_{et} {\Gamma^e}_{bi} - {\Gamma^a}_{ei} {\Gamma^e}_{bt} - {\Gamma^a}_{be} {c_{ti}}^e$
${R^a}_{bit} = -{R^a}_{bti}$
${R^a}_{bjk} = {\Gamma^a}_{bk,j} - {\Gamma^a}_{bj,k} + {\Gamma^a}_{ej} {\Gamma^e}_{bk} - {\Gamma^a}_{ek} {\Gamma^e}_{bj} - {\Gamma^a}_{be} {c_{jk}}^e$

Reformulate as vector field identity differentials:
$\gamma^{ij} {\Gamma^a}_{bt,ij} = \gamma^{ij} ({\Gamma^a}_{bi,t} - {R^a}_{bti} + {\Gamma^a}_{et} {\Gamma^e}_{bi} - {\Gamma^a}_{ei} {\Gamma^e}_{bt} - {\Gamma^a}_{be} {c_{ti}}^e)_{,j}$
$\epsilon^{ijk} {\Gamma^a}_{bk,j} = \epsilon^{ijk} ( {R^a}_{bjk} - {\Gamma^a}_{ej} {\Gamma^e}_{bk} + {\Gamma^a}_{ek} {\Gamma^e}_{bj} + {\Gamma^a}_{be} {c_{jk}}^e)$

Integrate divergence equation:
$\int {\Gamma^a}_{bt,ij} \gamma^{ij} dx^k = \int \gamma^{ij} ({\Gamma^a}_{bi,t} - {R^a}_{bti} + {\Gamma^a}_{et} {\Gamma^e}_{bi} - {\Gamma^a}_{ei} {\Gamma^e}_{bt} - {\Gamma^a}_{be} {c_{ti}}^e)_{,j} dx^k$
...to solve for ${\Gamma^a}_{bt}$, including ${\Gamma^a}_{tt}$, ${\Gamma^a}_{it}$

Integrate curl equation:
$\int \epsilon^{ijk} {\Gamma^a}_{bk,j} dx_k = \int \epsilon^{ijk} ({R^a}_{bjk} - {\Gamma^a}_{ej} {\Gamma^e}_{bk} + {\Gamma^a}_{ek} {\Gamma^e}_{bj} + {\Gamma^a}_{be} {c_{jk}}^e) dx_k$
...to solve for ${\Gamma^a}_{bk}$, including ${\Gamma^a}_{ti}$, ${\Gamma^a}_{ij}$

In the case that ${\Gamma^a}_{bc} = {\Gamma^a}_{(bc)}$ then we can solve for all of ${\Gamma^a}_{bc}$ with two methods to solve for ${\Gamma^a}_{ti}$.
If it is not symmetric then there is just enough solutions for all components of ${\Gamma^a}_{bc}$.