$\alpha$
${ \beta} ^i$
${{ \gamma} _i} _j$
${{ \gamma} ^i} ^j$
${{{ g} _u} _v} = {\overset{U\downarrow V\rightarrow}{\left[\begin{array}{cc} {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} _k}}}& { \beta} _j\\ { \beta} _i& {{ \gamma} _i} _j\end{array}\right]}}$
${{{ g} ^u} ^v} = {\overset{U\downarrow V\rightarrow}{\left[\begin{array}{cc} \frac{-1}{{\alpha}^{2}}& \frac{{ \beta} ^j}{{\alpha}^{2}}\\ \frac{{ \beta} ^i}{{\alpha}^{2}}& {{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}\end{array}\right]}}$
${{{{ \Gamma} _a} _b} _c} = {{\frac{1}{2}}{\left({{{{{{ g} _a} _b} _{,c}} + {{{{ g} _a} _c} _{,b}}}{-{{{{ g} _b} _c} _{,a}}}}\right)}}$