We use explicit expressions for ${\Gamma^{i}}_{kl}$
\begin{equation}\label{GammaGen}
\Gamma _{\alpha \beta }^{\mu }= \frac{1}{2}g^{\mu \nu }\left[
\frac{\partial g_{\alpha \nu }}{\partial x^{\beta }}+\frac{\partial g_{\beta \nu }}{\partial x^{\alpha }}-\frac{\partial g_{\alpha \beta }}{\partial x^{\nu }} \right].
\end{equation}
Here we use Greek alphabet for spacetime indices $\alpha,\beta,\nu=0,1,2,3$ and Latin for spatial indices $i,j=1,2,3$.
First, we consider the flat case. Then
\[\Gamma _{\alpha \beta }^{0} =-\frac{1}{2}
\frac{\partial {{g}_{\alpha \beta }}}{\partial t}.\]
The derivatives are non-zero only if $\alpha ,\beta $ are spatial indices $i,j=1,2,3$:
\[\Gamma _{00}^{0}=0,\quad \Gamma _{0i}^{0} =\Gamma _{i0}^{0}=0,\quad \Gamma _{ij}^{0} ={{\delta }_{ij}}\dot{a}a.\]
The symbols $\Gamma _{\alpha \beta }^{i}$ are non-zero only if one of the indices is spatial and the other is temporal
\[\Gamma _{0j}^{i}=\Gamma _{j0}^{i} ={{\delta }_{ij}}\frac{{\dot{a}}}{a}.\]
In the general case we present the FLRW metric in the form
\[d{{s}^{2}}=d{{t}^{2}}-{{a}^{2}}(t)
\gamma_{ij}(x)d{{x}^{i}}d{{x}^{j}},\]
where the non-zero components of the three-dimensional metric $\gamma_{ij}$ are
\[\gamma_{11}= \gamma_{rr} = \frac{1}{1-k r^2},\quad \gamma_{22}=\gamma_{\theta \theta}=r^2;\quad \gamma_{33} = \gamma_{\varphi \varphi } = r^2\sin^2\theta.\]
The non-zero components of the metric tensor ${{g}_{\alpha \beta }}$ and its inverse ${{g}^{\alpha \beta }}$ are the following:
\[ g_{00}= 1,~g_{ij}=- a^2(t)\gamma_{ij};
\quad g^{00}=1,
\quad g^{ij}=-\frac{1}{a^2(t)}{\gamma^{ij}}.\]
Using again the expression (\ref{GammaGen}), for the non-zero Christoffel symbols we obtain
\[ \Gamma _{0j}^{i} =\frac{1}{2}g^{ik}\frac{\partial g_{jk}}{\partial t} = \frac{\dot{a}}{a}\delta_{ij};
\quad\Gamma _{ij}^0=a\dot{a}{\gamma _{ij}}.\]
The symbols $\Gamma _{jk}^{i}$ are calculated also with the help of (\ref{GammaGen}), where instead of the metric tensor $g^{\mu\nu}$ one uses the three-dimensional metric $\gamma_{ij}$. Therefore all the non-zero Christoffel symbols generated by the FLRW-metric are
\begin{align*}
&\Gamma _{11}^{0}=\frac{a\dot{a}}{1-k{{r}^{2}}};\quad
\Gamma _{22}^{0}= a\dot{a}{r^2};\quad
\Gamma _{33}^{0}=a\dot{a}{r^2}\sin^2\theta\\
&\Gamma_{10}^1=\Gamma _{01}^1 =\frac{\dot a}{a};
\quad
\Gamma _{11}^1 =\frac{kr}{1-kr^2};\quad
\Gamma _{22}^1=-r(1-kr^2);\quad
\Gamma _{33}^1=-r(1-kr^2)\sin^2\theta;\\
&\Gamma^{2}_{20}=\Gamma^{2}_{02} = \frac{\dot{a}}{a};\quad
\Gamma^{2}_{21}=\Gamma^{2}_{12}=\frac 1 r ;\quad
\Gamma^{2}_{33}=-\sin\theta\cos\theta;\\
&\Gamma^{3}_{30}=\Gamma^{3}_{03} =\frac{\dot{a}}{a};\quad
\Gamma^{3}_{31}=\Gamma^{3}_{13}=\frac 1 r ;\quad
\Gamma^{3}_{32}=\Gamma^{3}_{23} =\cot\theta.
\end{align*}
