cylindrical, anholonomic, conformal
chart coordinates: $x^\tilde{\mu} = \{r, \theta, z\}$
chart coordinate basis: $e_\tilde{\mu} = \{e_{\tilde{r}}, e_{\tilde{\theta}}, e_{\tilde{z}}\}$
embedding coordinates: $u^I = \{x, y, z\}$
embedding basis $e_I = \{e_{x}, e_{y}, e_{z}\}$
flat metric:
${{{ \eta} _I} _J} = {\overset{I\downarrow J\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix} \right]}}$
transform from basis to coordinate:
${{{ \tilde{e}} _A} ^a} = {\overset{a\downarrow A\rightarrow}{\left[ \begin{matrix} {r}^{\frac{1}{3}} & 0 & 0 \\ 0 & {r}^{\frac{1}{3}} & 0 \\ 0 & 0 & {r}^{\frac{1}{3}}\end{matrix} \right]}}$
transform from coorinate to basis:
${{{ \tilde{e}} ^a} _A} = {\overset{a\downarrow A\rightarrow}{\left[ \begin{matrix} \frac{1}{\sqrt[3]{r}} & 0 & 0 \\ 0 & \frac{1}{\sqrt[3]{r}} & 0 \\ 0 & 0 & \frac{1}{\sqrt[3]{r}}\end{matrix} \right]}}$
tensor index associated with coordinate $r$ has operator $e_{r}(\zeta) = $$\frac{\frac{\partial \zeta}{\partial r}}{\sqrt[3]{r}}$
tensor index associated with coordinate $\theta$ has operator $e_{\theta}(\zeta) = $$\frac{\frac{\partial \zeta}{\partial \theta}}{\sqrt[3]{r}}$
tensor index associated with coordinate $z$ has operator $e_{z}(\zeta) = $$\frac{\frac{\partial \zeta}{\partial z}}{\sqrt[3]{r}}$
chart in embedded coordinates:
${u} = {\overset{I\downarrow}{\left[ \begin{matrix} {{r}} {{\cos\left( \theta\right)}} \\ {{r}} {{\sin\left( \theta\right)}} \\ z\end{matrix} \right]}}$
basis operators applied to chart:
${{{ e} _u} ^I} = {{{ u} ^I} _{,u}}$
${{{ e} _u} ^I} = {\overset{u\downarrow I\rightarrow}{\left[ \begin{matrix} \frac{\cos\left( \theta\right)}{\sqrt[3]{r}} & \frac{\sin\left( \theta\right)}{\sqrt[3]{r}} & 0 \\ -{{{\sqrt[3]{{r}^{2}}}} {{\sin\left( \theta\right)}}} & {{\sqrt[3]{{r}^{2}}}} {{\cos\left( \theta\right)}} & 0 \\ 0 & 0 & \frac{1}{\sqrt[3]{r}}\end{matrix} \right]}}$
${{{ e} ^u} _I} = {\overset{u\downarrow I\rightarrow}{\left[ \begin{matrix} {{\sqrt[3]{r}}} {{\cos\left( \theta\right)}} & {{\sqrt[3]{r}}} {{\sin\left( \theta\right)}} & 0 \\ -{\frac{\sin\left( \theta\right)}{\sqrt[3]{{r}^{2}}}} & \frac{\cos\left( \theta\right)}{\sqrt[3]{{r}^{2}}} & 0 \\ 0 & 0 & \sqrt[3]{r}\end{matrix} \right]}}$
${{{{{ e} _u} ^I}} {{{{ e} ^v} _I}}} = {\overset{u\downarrow v\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix} \right]}}$
${{{{{ e} _u} ^I}} {{{{ e} ^u} _J}}} = {\overset{I\downarrow J\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix} \right]}}$
basis determinant: ${det(e)} = {1}$
${{{{ c} _a} _b} ^c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & -{\frac{1}{{{3}} {{{{r}} {{\sqrt[3]{r}}}}}}} & 0 \\ 0 & 0 & -{\frac{1}{{{3}} {{{{r}} {{\sqrt[3]{r}}}}}}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & \frac{1}{{{3}} {{{{r}} {{\sqrt[3]{r}}}}}} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & \frac{1}{{{3}} {{{{r}} {{\sqrt[3]{r}}}}}} \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
${{{ g} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[ \begin{matrix} \frac{1}{\sqrt[3]{{r}^{2}}} & 0 & 0 \\ 0 & {{r}} {{\sqrt[3]{r}}} & 0 \\ 0 & 0 & \frac{1}{\sqrt[3]{{r}^{2}}}\end{matrix} \right]}}$
${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
metric determinant: ${det(g)} = {1}$
${{{{ \Gamma} _a} _b} _c} = {{{\frac{1}{2}}} {{\left({{{{{{{ g} _a} _b} _{,c}} + {{{{ g} _a} _c} _{,b}}}{-{{{{ g} _b} _c} _{,a}}} + {{{{ c} _a} _b} _c} + {{{{ c} _a} _c} _b}}{-{{{{ c} _c} _b} _a}}}\right)}}}$
commutation coefficients:
${c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & -{\frac{1}{{{3}} {{{{r}} {{\sqrt[3]{r}}}}}}} & 0 \\ 0 & 0 & -{\frac{1}{{{3}} {{{{r}} {{\sqrt[3]{r}}}}}}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & \frac{1}{{{3}} {{{{r}} {{\sqrt[3]{r}}}}}} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & \frac{1}{{{3}} {{{{r}} {{\sqrt[3]{r}}}}}} \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
metric:
${g} = {\overset{u\downarrow v\rightarrow}{\left[ \begin{matrix} \frac{1}{\sqrt[3]{{r}^{2}}} & 0 & 0 \\ 0 & {{r}} {{\sqrt[3]{r}}} & 0 \\ 0 & 0 & \frac{1}{\sqrt[3]{{r}^{2}}}\end{matrix} \right]}}$
metric inverse:
${g} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \sqrt[3]{{r}^{2}} & 0 & 0 \\ 0 & \frac{1}{{{r}} {{\sqrt[3]{r}}}} & 0 \\ 0 & 0 & \sqrt[3]{{r}^{2}}\end{matrix} \right]}}$
metric derivative:
${{\partial g}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} -{\frac{2}{{{3}} {{{r}^{2}}}}} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ \frac{4}{3} & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ -{\frac{2}{{{3}} {{{r}^{2}}}}} & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
1st kind Christoffel:
${\Gamma} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} -{\frac{1}{{{3}} {{{r}^{2}}}}} & 0 & 0 \\ 0 & -{1} & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & \frac{2}{3} & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & -{\frac{1}{{{3}} {{{r}^{2}}}}} \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
connection coefficients / 2nd kind Christoffel:
${\Gamma} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} -{\frac{1}{{{3}} {{{{r}} {{\sqrt[3]{r}}}}}}} & 0 & 0 \\ 0 & -{\sqrt[3]{{r}^{2}}} & 0 \\ 0 & 0 & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & \frac{2}{{{3}} {{{{r}} {{\sqrt[3]{r}}}}}} & 0 \\ \frac{1}{{{r}} {{\sqrt[3]{r}}}} & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & -{\frac{1}{{{3}} {{{{r}} {{\sqrt[3]{r}}}}}}} \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]\end{matrix} \right]}}$
connection coefficients derivative:
${{\partial \Gamma}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{4}{{{9}} {{{{{r}^{2}}} {{\sqrt[3]{{r}^{2}}}}}}} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ -{\frac{2}{{{3}} {{\sqrt[3]{{r}^{2}}}}}} & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ -{\frac{8}{{{9}} {{{{{r}^{2}}} {{\sqrt[3]{{r}^{2}}}}}}}} & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} -{\frac{4}{{{3}} {{{{{r}^{2}}} {{\sqrt[3]{{r}^{2}}}}}}}} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ \frac{4}{{{9}} {{{{{r}^{2}}} {{\sqrt[3]{{r}^{2}}}}}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
connection coefficients squared:
${{(\Gamma^2)}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{1}{{{9}} {{{{{r}^{2}}} {{\sqrt[3]{{r}^{2}}}}}}} & 0 & 0 \\ 0 & -{\frac{1}{\sqrt[3]{{r}^{2}}}} & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{1}{{{3}} {{\sqrt[3]{{r}^{2}}}}} & 0 \\ -{\frac{2}{{{3}} {{\sqrt[3]{{r}^{2}}}}}} & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{2}{{{3}} {{{{{r}^{2}}} {{\sqrt[3]{{r}^{2}}}}}}} & 0 \\ -{\frac{1}{{{3}} {{{{{r}^{2}}} {{\sqrt[3]{{r}^{2}}}}}}}} & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{4}{{{9}} {{{{{r}^{2}}} {{\sqrt[3]{{r}^{2}}}}}}} & 0 & 0 \\ 0 & -{\frac{1}{\sqrt[3]{{r}^{2}}}} & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{1}{{{9}} {{{{{r}^{2}}} {{\sqrt[3]{{r}^{2}}}}}}} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${R} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${R} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]\end{matrix} \right]}}$
Ricci curvature, $\sharp\flat$:
${R} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}}$
Gaussian curvature:
$0$
trace-free Ricci, $\sharp\flat$:
${{(R^{TF})}} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${G} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}}$
Schouten, $\sharp\flat$:
${P} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}}$
Weyl, $\sharp\sharp\flat\flat$:
${C} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
Weyl, $\flat\flat\flat\flat$:
${C} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]\end{matrix} \right]}}$
Plebanski, $\sharp\sharp\flat\flat$:
${P} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
divergence: ${{{{ A} ^i} _{,i}} + {{{{{{ \Gamma} ^i} _i} _j}} {{{ A} ^j}}}} = {{{{2}} \cdot {{\frac{1}{3}}} {{{A^{\hat{r}}}}} \cdot {{\frac{1}{r}}} {{\frac{1}{\sqrt[3]{r}}}}} + {{{\frac{1}{\sqrt[3]{r}}}} {{\frac{\partial {A^{\hat{\theta}}}}{\partial \theta}}}} + {{{\frac{1}{\sqrt[3]{r}}}} {{\frac{\partial {A^{\hat{r}}}}{\partial r}}}} + {{{\frac{1}{\sqrt[3]{r}}}} {{\frac{\partial {A^{\hat{z}}}}{\partial z}}}}}$
geodesic:
${\overset{a\downarrow}{\left[ \begin{matrix} \ddot{\hat{r}} \\ \ddot{\hat{\theta}} \\ \ddot{\hat{z}}\end{matrix} \right]}} = {\overset{a\downarrow}{\left[ \begin{matrix} {{{\frac{1}{3}}} {{{\dot{\hat{r}}}^{2}}} {{\frac{1}{r}}} {{\frac{1}{\sqrt[3]{r}}}}} + {{{{\dot{\hat{\theta}}}^{2}}} {{\sqrt[3]{{r}^{2}}}}} \\ {{-5}} \cdot {{\frac{1}{3}}} {{\dot{\hat{\theta}}}} \cdot {{\dot{\hat{r}}}} \cdot {{\frac{1}{{r}^{{{4}} \cdot {{\frac{1}{3}}}}}}} \\ {{\frac{1}{3}}} {{\dot{\hat{r}}}} \cdot {{\dot{\hat{z}}}} \cdot {{\frac{1}{{r}^{{{4}} \cdot {{\frac{1}{3}}}}}}}\end{matrix} \right]}}$
parallel propagators:
${{[\Gamma_r]}} = {\left[ \begin{matrix} -{\frac{1}{{{3}} {{r}}}} & 0 & 0 \\ 0 & \frac{2}{{{3}} {{r}}} & 0 \\ 0 & 0 & -{\frac{1}{{{3}} {{r}}}}\end{matrix} \right]}$
$\int\limits_{{{r_L}}}^{{{r_R}}} {\left[ \begin{matrix} -{\frac{1}{{{3}} {{r}}}} & 0 & 0 \\ 0 & \frac{2}{{{3}} {{r}}} & 0 \\ 0 & 0 & -{\frac{1}{{{3}} {{r}}}}\end{matrix} \right]}d r$
= $\left[ \begin{matrix} \log\left( {\sqrt[3]{{\frac{1}{{r_R}}} {{r_L}}}}\right) & 0 & 0 \\ 0 & \log\left( {\frac{\sqrt[3]{{{r_R}}^{2}}}{\sqrt[3]{{{r_L}}^{2}}}}\right) & 0 \\ 0 & 0 & \log\left( {\sqrt[3]{{\frac{1}{{r_R}}} {{r_L}}}}\right)\end{matrix} \right]$
${ P} _r$ = $\exp\left( -{\int\limits_{{{r_L}}}^{{{r_R}}} {\left[ \begin{matrix} -{\frac{1}{{{3}} {{r}}}} & 0 & 0 \\ 0 & \frac{2}{{{3}} {{r}}} & 0 \\ 0 & 0 & -{\frac{1}{{{3}} {{r}}}}\end{matrix} \right]}d r}\right)$