metric:
${{{ g} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} -{1}& 0& 0& 0\\ 0& \frac{{a}^{2}}{{1}{-{{{k}} {{{r}^{2}}}}}}& 0& 0\\ 0& 0& {{{a}^{2}}} {{{r}^{2}}}& 0\\ 0& 0& 0& {{{a}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}\end{array}\right]}}$
metric inverse:
${{{ g} ^u} ^v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} -{1}& 0& 0& 0\\ 0& \frac{{1}{-{{{k}} {{{r}^{2}}}}}}{{a}^{2}}& 0& 0\\ 0& 0& \frac{1}{{{{a}^{2}}} {{{r}^{2}}}}& 0\\ 0& 0& 0& \frac{1}{{{{a}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}\end{array}\right]}}$
1st kind Christoffel:
${{{{ \Gamma} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{{{a}} {{\frac{\partial a}{\partial t}}}}{{1}{-{{{k}} {{{r}^{2}}}}}}}& 0& 0\\ 0& 0& -{{{a}} {{{r}^{2}}} {{\frac{\partial a}{\partial t}}}}& 0\\ 0& 0& 0& {{a}} {{\frac{\partial a}{\partial t}}} {{\left({{-{{r}^{2}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{a}} {{\frac{\partial a}{\partial t}}}}{{1}{-{{{k}} {{{r}^{2}}}}}}& 0& 0\\ \frac{{{a}} {{\frac{\partial a}{\partial t}}}}{{1}{-{{{k}} {{{r}^{2}}}}}}& \frac{{{k}} {{r}} {{{a}^{2}}}}{{1} + {{{{k}^{2}}} {{{r}^{4}}}}{-{{{2}} {{k}} {{{r}^{2}}}}}}& 0& 0\\ 0& 0& -{{{r}} {{{a}^{2}}}}& 0\\ 0& 0& 0& {{r}} {{\left({{-{{a}^{2}}} + {{{{a}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& {{a}} {{{r}^{2}}} {{\frac{\partial a}{\partial t}}}& 0\\ 0& 0& {{r}} {{{a}^{2}}}& 0\\ {{a}} {{{r}^{2}}} {{\frac{\partial a}{\partial t}}}& {{r}} {{{a}^{2}}}& 0& 0\\ 0& 0& 0& -{{{{a}^{2}}} {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& {{a}} {{\frac{\partial a}{\partial t}}} {{\left({{{r}^{2}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}\\ 0& 0& 0& {{r}} {{\left({{{a}^{2}}{-{{{{a}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}\\ 0& 0& 0& {{{a}^{2}}} {{{r}^{2}}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}}\\ {{a}} {{\frac{\partial a}{\partial t}}} {{\left({{{r}^{2}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}& {{r}} {{\left({{{a}^{2}}{-{{{{a}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}& {{{a}^{2}}} {{{r}^{2}}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}}& 0\end{array}\right]}\end{matrix}\right]}}$

2nd kind Christoffel:
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \frac{{{a}} {{\frac{\partial a}{\partial t}}}}{{1}{-{{{k}} {{{r}^{2}}}}}}& 0& 0\\ 0& 0& {{a}} {{{r}^{2}}} {{\frac{\partial a}{\partial t}}}& 0\\ 0& 0& 0& {{a}} {{\frac{\partial a}{\partial t}}} {{\left({{{r}^{2}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}\end{array}\right] \\ \left[\begin{array}{cccc} 0& {\frac{1}{a}} {\frac{\partial a}{\partial t}}& 0& 0\\ {\frac{1}{a}} {\frac{\partial a}{\partial t}}& \frac{{{k}} {{r}} {{\left({{1}{-{{{k}} {{{r}^{2}}}}}}\right)}}}{{1} + {{{{k}^{2}}} {{{r}^{4}}}}{-{{{2}} {{k}} {{{r}^{2}}}}}}& 0& 0\\ 0& 0& {{r}} {{\left({{-{1}} + {{{k}} {{{r}^{2}}}}}\right)}}& 0\\ 0& 0& 0& {{r}} {{\left({{-{1}} + {{{k}} {{{r}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{k}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& {\frac{1}{a}} {\frac{\partial a}{\partial t}}& 0\\ 0& 0& \frac{1}{r}& 0\\ {\frac{1}{a}} {\frac{\partial a}{\partial t}}& \frac{1}{r}& 0& 0\\ 0& 0& 0& -{{{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}}}\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& {\frac{1}{a}} {\frac{\partial a}{\partial t}}\\ 0& 0& 0& \frac{1}{r}\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}\\ {\frac{1}{a}} {\frac{\partial a}{\partial t}}& \frac{1}{r}& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}& 0\end{array}\right]\end{matrix}\right]}}$

geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} \frac{{\ddot{t}}{-{{{\ddot{t}}} \cdot {{k}} {{{r}^{2}}}}} + {{{a}} {{{\dot{r}}^{2}}} {{\frac{\partial a}{\partial t}}}} + {{{a}} {{{\dot{\phi}}^{2}}} {{{r}^{2}}} {{\frac{\partial a}{\partial t}}}} + {{{a}} {{{\dot{\theta}}^{2}}} {{{r}^{2}}} {{\frac{\partial a}{\partial t}}}}{-{{{a}} {{{\dot{\phi}}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial a}{\partial t}}}}}{-{{{a}} {{k}} {{{\dot{\phi}}^{2}}} {{{r}^{4}}} {{\frac{\partial a}{\partial t}}}}}{-{{{a}} {{k}} {{{\dot{\theta}}^{2}}} {{{r}^{4}}} {{\frac{\partial a}{\partial t}}}}} + {{{a}} {{k}} {{{\dot{\phi}}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial a}{\partial t}}}}}{{1}{-{{{k}} {{{r}^{2}}}}}} \\ \frac{{{{\ddot{r}}} \cdot {{a}}}{-{{{a}} {{r}} {{{\dot{\phi}}^{2}}}}}{-{{{a}} {{r}} {{{\dot{\theta}}^{2}}}}} + {{{a}} {{r}} {{{\dot{\phi}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{a}} {{{\dot{r}}^{2}}} {{{k}^{2}}} {{{r}^{3}}}}} + {{{2}} {{\dot{r}}} \cdot {{\dot{t}}} \cdot {{\frac{\partial a}{\partial t}}}}{-{{{3}} {{a}} {{{\dot{\phi}}^{2}}} {{{k}^{2}}} {{{r}^{5}}}}}{-{{{3}} {{a}} {{{\dot{\theta}}^{2}}} {{{k}^{2}}} {{{r}^{5}}}}} + {{{\ddot{r}}} \cdot {{a}} {{{k}^{2}}} {{{r}^{4}}}} + {{{a}} {{k}} {{r}} {{{\dot{r}}^{2}}}} + {{{a}} {{{\dot{\phi}}^{2}}} {{{k}^{3}}} {{{r}^{7}}}} + {{{a}} {{{\dot{\theta}}^{2}}} {{{k}^{3}}} {{{r}^{7}}}}{-{{{a}} {{{\dot{\phi}}^{2}}} {{{k}^{3}}} {{{r}^{7}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\ddot{r}}} \cdot {{a}} {{k}} {{{r}^{2}}}}} + {{{3}} {{a}} {{k}} {{{\dot{\phi}}^{2}}} {{{r}^{3}}}} + {{{3}} {{a}} {{k}} {{{\dot{\theta}}^{2}}} {{{r}^{3}}}}{-{{{3}} {{a}} {{k}} {{{\dot{\phi}}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{3}} {{a}} {{{\dot{\phi}}^{2}}} {{{k}^{2}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\dot{r}}} \cdot {{\dot{t}}} \cdot {{{k}^{2}}} {{{r}^{4}}} {{\frac{\partial a}{\partial t}}}}{-{{{4}} {{\dot{r}}} \cdot {{\dot{t}}} \cdot {{k}} {{{r}^{2}}} {{\frac{\partial a}{\partial t}}}}}}{{{a}} {{\left({{1} + {{{{k}^{2}}} {{{r}^{4}}}}{-{{{2}} {{k}} {{{r}^{2}}}}}}\right)}}} \\ \frac{{{{\ddot{\theta}}} \cdot {{a}} {{r}}} + {{{2}} {{\dot{\theta}}} \cdot {{\dot{r}}} \cdot {{a}}} + {{{2}} {{\dot{\theta}}} \cdot {{\dot{t}}} \cdot {{r}} {{\frac{\partial a}{\partial t}}}}{-{{{a}} {{r}} {{{\dot{\phi}}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}}{{{a}} {{r}}} \\ \frac{{{{\ddot{\phi}}} \cdot {{a}} {{r}} {{\sin\left( \theta\right)}}} + {{{2}} {{\dot{\phi}}} \cdot {{\dot{r}}} \cdot {{a}} {{\sin\left( \theta\right)}}} + {{{2}} {{\dot{\phi}}} \cdot {{\dot{\theta}}} \cdot {{a}} {{r}} {{\cos\left( \theta\right)}}} + {{{2}} {{\dot{\phi}}} \cdot {{\dot{t}}} \cdot {{r}} {{\frac{\partial a}{\partial t}}} {{\sin\left( \theta\right)}}}}{{{a}} {{r}} {{\sin\left( \theta\right)}}}\end{matrix}\right]}} = {\overset{a\downarrow}{\left[\begin{matrix} 0 \\ 0 \\ 0 \\ 0\end{matrix}\right]}}$

Riemann curvature tensor:
${{{{{ R} ^a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& \frac{{{a}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}{{1}{-{{{k}} {{{r}^{2}}}}}}& 0& 0\\ -{\frac{{{a}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}{{1}{-{{{k}} {{{r}^{2}}}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& {{a}} {{{r}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}& 0\\ 0& 0& 0& 0\\ -{{{a}} {{{r}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& {{a}} {{\frac{\partial^ 2 a}{\partial t^ 2}}} {{\left({{{r}^{2}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ {{a}} {{\frac{\partial^ 2 a}{\partial t^ 2}}} {{\left({{-{{r}^{2}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}}& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& \frac{{{\frac{\partial^ 2 a}{\partial t^ 2}}} {{\left({{-{1}} + {{{k}} {{{r}^{2}}}}}\right)}}}{{a}^{3}}& 0& 0\\ \frac{{{\frac{\partial^ 2 a}{\partial t^ 2}}} {{\left({{1}{-{{{k}} {{{r}^{2}}}}}}\right)}}}{{a}^{3}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{-{{{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}}{-{{{k}} {{{r}^{2}}}}} + {{{{k}^{2}}} {{{r}^{4}}}} + {{{k}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}}}}{{a}^{2}}& 0\\ 0& \frac{{{{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}} + {{{k}} {{{r}^{2}}}}{-{{{{k}^{2}}} {{{r}^{4}}}}}{-{{{k}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}}}}}{{a}^{2}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}}{-{{{k}} {{{r}^{2}}}}} + {{{k}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{k}^{2}}} {{{r}^{4}}}}{-{{{{k}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{k}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}}}{-{{{k}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}{{a}^{2}}\\ 0& 0& 0& 0\\ 0& \frac{{{{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}} + {{{k}} {{{r}^{2}}}}{-{{{k}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{k}^{2}}} {{{r}^{4}}}}} + {{{{k}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{k}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}}}} + {{{k}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{{a}^{2}}& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& -{\frac{\frac{\partial^ 2 a}{\partial t^ 2}}{{{{a}^{3}}} {{{r}^{2}}}}}& 0\\ 0& 0& 0& 0\\ \frac{\frac{\partial^ 2 a}{\partial t^ 2}}{{{{a}^{3}}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{{k}^{3}} + {{\frac{\partial a}{\partial t}}^{6}} + {{{3}} {{{k}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}} + {{{3}} {{k}} {{{\frac{\partial a}{\partial t}}^{4}}}}}{{{{{a}^{2}}} {{{k}^{2}}} {{{r}^{2}}}} + {{{{a}^{2}}} {{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}}}{-{{{k}} {{{a}^{2}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{4}}}}}{-{{{{a}^{2}}} {{{k}^{3}}} {{{r}^{4}}}}} + {{{2}} {{k}} {{{a}^{2}}} {{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}{-{{{2}} {{{a}^{2}}} {{{k}^{2}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}}}}}& 0\\ 0& \frac{{{{k}^{2}}} {{\left({{{k}^{2}} + {{\frac{\partial a}{\partial t}}^{4}} + {{{2}} {{k}} {{{\frac{\partial a}{\partial t}}^{2}}}}}\right)}}}{{{{{a}^{2}}} {{{k}^{4}}} {{{r}^{4}}}}{-{{{{a}^{2}}} {{{k}^{3}}} {{{r}^{2}}}}}{-{{{{a}^{2}}} {{{k}^{2}}} {{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}} + {{{{a}^{2}}} {{{k}^{3}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{k}}{-{{\frac{\partial a}{\partial t}}^{2}}} + {{{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}} + {{{k}} {{{\cos\left( \theta\right)}^{2}}}}}{{a}^{2}}\\ 0& 0& \frac{{k} + {{\frac{\partial a}{\partial t}}^{2}}{-{{{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{k}} {{{\cos\left( \theta\right)}^{2}}}}}}{{a}^{2}}& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& -{\frac{\frac{\partial^ 2 a}{\partial t^ 2}}{{{{a}^{3}}} {{\left({{{r}^{2}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{\frac{\partial^ 2 a}{\partial t^ 2}}{{{{a}^{3}}} {{\left({{{r}^{2}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)}}}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{k}^{3}} + {{\frac{\partial a}{\partial t}}^{6}} + {{{3}} {{{k}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}} + {{{3}} {{k}} {{{\frac{\partial a}{\partial t}}^{4}}}}}{{{{{a}^{2}}} {{{k}^{2}}} {{{r}^{2}}}} + {{{{a}^{2}}} {{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}}}{-{{{k}} {{{a}^{2}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{4}}}}}{-{{{{a}^{2}}} {{{k}^{3}}} {{{r}^{4}}}}}{-{{{{a}^{2}}} {{{k}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{a}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}}}} + {{{{a}^{2}}} {{{k}^{3}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{k}} {{{a}^{2}}} {{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}{-{{{2}} {{{a}^{2}}} {{{k}^{2}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}}}} + {{{k}} {{{a}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}}}{-{{{2}} {{k}} {{{a}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}} + {{{2}} {{{a}^{2}}} {{{k}^{2}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}\\ 0& 0& 0& 0\\ 0& \frac{{{{k}^{2}}} {{\left({{{k}^{2}} + {{\frac{\partial a}{\partial t}}^{4}} + {{{2}} {{k}} {{{\frac{\partial a}{\partial t}}^{2}}}}}\right)}}}{{{{{a}^{2}}} {{{k}^{4}}} {{{r}^{4}}}}{-{{{{a}^{2}}} {{{k}^{3}}} {{{r}^{2}}}}}{-{{{{a}^{2}}} {{{k}^{2}}} {{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}} + {{{{a}^{2}}} {{{k}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{a}^{2}}} {{{k}^{3}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}}}{-{{{{a}^{2}}} {{{k}^{4}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{a}^{2}}} {{{k}^{2}}} {{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{a}^{2}}} {{{k}^{3}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{k} + {{\frac{\partial a}{\partial t}}^{2}}{-{{{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{k}} {{{\cos\left( \theta\right)}^{2}}}}}}{{{{a}^{2}}} {{{\sin\left( \theta\right)}^{4}}}}\\ 0& 0& \frac{{-{k}}{-{{\frac{\partial a}{\partial t}}^{2}}} + {{{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}} + {{{k}} {{{\cos\left( \theta\right)}^{2}}}}}{{{{a}^{2}}} {{{\sin\left( \theta\right)}^{4}}}}& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{array}\right]}}$
Ricci curvature tensor:
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{{\left({{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{r}^{4}} + {{{2}} {{{r}^{2}}}}{-{{{k}} {{{r}^{6}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{k}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}{{{{a}^{3}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\\ 0& \frac{{-{{{2}} {{{a}^{2}}} {{{k}^{7}}} {{{r}^{2}}}}} + {{{{a}^{2}}} {{{k}^{7}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{a}^{2}}} {{{k}^{9}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{a}^{5}}} {{{k}^{8}}} {{{r}^{8}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}{-{{{2}} {{{a}^{2}}} {{{k}^{6}}} {{{r}^{6}}} {{{\frac{\partial a}{\partial t}}^{6}}}}}{-{{{2}} {{{a}^{2}}} {{{k}^{9}}} {{{r}^{6}}}}}{-{{{2}} {{{a}^{2}}} {{{k}^{4}}} {{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{6}}}}}{-{{{2}} {{{a}^{2}}} {{{k}^{8}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{4}} {{{a}^{2}}} {{{k}^{8}}} {{{r}^{4}}}} + {{{{a}^{2}}} {{{k}^{4}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{6}}}} + {{{{a}^{2}}} {{{k}^{6}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{6}}}} + {{{{a}^{5}}} {{{k}^{4}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{4}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}} + {{{{a}^{5}}} {{{k}^{6}}} {{{r}^{4}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}{-{{{{a}^{5}}} {{{k}^{8}}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}}{-{{{6}} {{{a}^{2}}} {{{k}^{5}}} {{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}}}}{-{{{6}} {{{a}^{2}}} {{{k}^{6}}} {{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}}{-{{{6}} {{{a}^{2}}} {{{k}^{8}}} {{{r}^{6}}} {{{\frac{\partial a}{\partial t}}^{2}}}}}{-{{{2}} {{{a}^{2}}} {{{k}^{5}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{6}}}}}{-{{{2}} {{{a}^{5}}} {{{k}^{7}}} {{{r}^{6}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}} + {{{2}} {{{a}^{5}}} {{{k}^{5}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}{-{{{2}} {{{a}^{5}}} {{{k}^{5}}} {{{r}^{6}}} {{{\frac{\partial a}{\partial t}}^{4}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}} + {{{3}} {{{a}^{2}}} {{{k}^{5}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}}} + {{{3}} {{{a}^{2}}} {{{k}^{6}}} {{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{a}^{2}}} {{{k}^{8}}} {{{r}^{6}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{a}^{2}}} {{{k}^{7}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}}} + {{{12}} {{{a}^{2}}} {{{k}^{6}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{4}}}} + {{{12}} {{{a}^{2}}} {{{k}^{7}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}}} + {{{4}} {{{a}^{2}}} {{{k}^{5}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{6}}}}{-{{{4}} {{{a}^{5}}} {{{k}^{6}}} {{{r}^{6}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}}{-{{{{a}^{5}}} {{{k}^{4}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}}{-{{{{a}^{5}}} {{{k}^{6}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}} + {{{{a}^{5}}} {{{k}^{6}}} {{{r}^{8}}} {{{\frac{\partial a}{\partial t}}^{4}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}{-{{{6}} {{{a}^{2}}} {{{k}^{7}}} {{{r}^{6}}} {{{\frac{\partial a}{\partial t}}^{4}}}}}{-{{{6}} {{{a}^{2}}} {{{k}^{7}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}}{-{{{6}} {{{a}^{2}}} {{{k}^{6}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}}}} + {{{2}} {{{a}^{5}}} {{{k}^{7}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}} + {{{2}} {{{a}^{5}}} {{{k}^{7}}} {{{r}^{8}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}{-{{{2}} {{{a}^{5}}} {{{k}^{7}}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}}{-{{{2}} {{{a}^{5}}} {{{k}^{5}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}} + {{{2}} {{{a}^{5}}} {{{k}^{5}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}} + {{{4}} {{{a}^{5}}} {{{k}^{6}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}{-{{{{a}^{5}}} {{{k}^{6}}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}}}{{{{{a}^{4}}} {{{k}^{6}}} {{{r}^{4}}}} + {{{3}} {{{a}^{4}}} {{{k}^{8}}} {{{r}^{8}}}}{-{{{{a}^{4}}} {{{k}^{9}}} {{{r}^{10}}}}} + {{{{r}^{4}}} {{{a}^{4}}} {{{k}^{4}}} {{{\frac{\partial a}{\partial t}}^{4}}}}{-{{{{a}^{4}}} {{{k}^{6}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{a}^{4}}} {{{k}^{7}}} {{{r}^{10}}} {{{\frac{\partial a}{\partial t}}^{4}}}}} + {{{2}} {{{a}^{4}}} {{{k}^{5}}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}}}{-{{{3}} {{{a}^{4}}} {{{k}^{7}}} {{{r}^{6}}}}}{-{{{3}} {{{a}^{4}}} {{{k}^{8}}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{a}^{4}}} {{{k}^{5}}} {{{r}^{6}}} {{{\frac{\partial a}{\partial t}}^{4}}}}} + {{{3}} {{{a}^{4}}} {{{k}^{6}}} {{{r}^{8}}} {{{\frac{\partial a}{\partial t}}^{4}}}} + {{{{a}^{4}}} {{{k}^{9}}} {{{r}^{10}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{a}^{4}}} {{{k}^{4}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}}}} + {{{{k}^{7}}} {{{a}^{4}}} {{{r}^{10}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}}}{-{{{6}} {{{a}^{4}}} {{{k}^{6}}} {{{r}^{6}}} {{{\frac{\partial a}{\partial t}}^{2}}}}}{-{{{2}} {{{a}^{4}}} {{{k}^{8}}} {{{r}^{10}}} {{{\frac{\partial a}{\partial t}}^{2}}}}} + {{{2}} {{{a}^{4}}} {{{k}^{8}}} {{{r}^{10}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{2}} {{{a}^{4}}} {{{k}^{5}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}} + {{{3}} {{{a}^{4}}} {{{k}^{7}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{a}^{4}}} {{{k}^{5}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}}}{-{{{3}} {{{a}^{4}}} {{{k}^{6}}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{4}}}}} + {{{6}} {{{a}^{4}}} {{{k}^{6}}} {{{r}^{6}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}} + {{{6}} {{{a}^{4}}} {{{k}^{7}}} {{{r}^{8}}} {{{\frac{\partial a}{\partial t}}^{2}}}}{-{{{6}} {{{a}^{4}}} {{{k}^{7}}} {{{r}^{8}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}}}& 0& 0\\ 0& 0& \frac{{-{k}}{-{{\frac{\partial a}{\partial t}}^{2}}} + {{{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}{-{{{k}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{k}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{k}^{2}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{k}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{k}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}{-{{{{k}^{2}}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\sin\left( \theta\right)}^{2}}}} + {{{{a}^{3}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}{-{{{k}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}}{-{{{{a}^{3}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}}}{{{{a}^{2}}} {{{\sin\left( \theta\right)}^{4}}}}& 0\\ 0& 0& 0& \frac{{-{k}}{-{{\frac{\partial a}{\partial t}}^{2}}}{-{{{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}} + {{{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}}}{-{{{k}} {{{r}^{2}}}}} + {{{k}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{k}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{k}^{2}}} {{{r}^{4}}}}{-{{{{k}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{r}^{2}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{k}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}}}{-{{{k}} {{{r}^{4}}} {{{\frac{\partial a}{\partial t}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{a}^{3}}} {{{r}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}{-{{{{a}^{3}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 a}{\partial t^ 2}}}}}}{{a}^{2}}\end{array}\right]}}$
Gaussian curvature