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