The components of the gamma_[a, b, c] tensor are the Ricci rotation coefficients and the components of the lambda_[a, b, c] tensor are linear combinations of them, according to the definitions in the Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2 (definitions (98.9) and (98.10)). (See references). Thus,

where ${▿}_{\mathrm{\nu }}\left({𝔢}_{a,\mathrm{\mu }}\right)$, represented in Maple by D_[nu](e_[a,mu]), is the covariant derivative of the tetrad, and

From these definitions it follows that ${\mathrm{gamma}}_{a\,b\,c}$ is antisymmetric in its first pair of indices, and the covariant derivatives entering the definition of ${\mathrm{lambda}}_{a\,b\,c}$ can be replaced by non-covariant ones using the d_ operator. You can retrieve these definitions directly in the worksheet entering gamma_[definition] and lambda_[definition].

Both gamma_ and lambda_ accept the keywords accepted by the other tensors of the Physics package, these are definition, matrix and nonzero, that can be given with or without indices. If given with indices, the corresponding output takes their character (covariant or contravariant) into account.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$$\mathrm{with}\left(\mathrm{Tetrads}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Setting}\mathrm{lowercaselatin\_ah}\mathrm{letters\; to\; represent}\mathrm{tetrad}\mathrm{indices}$

$\mathrm{Defined\; as\; tetrad\; tensors}\left(\mathrm{see\; ?Physics,tetrads}\right),{𝔢}_{a,\mathrm{\mu }},{\mathrm{\eta }}_{a,b},{\mathrm{\gamma }}_{a,b,c},{\mathrm{\lambda }}_{a,b,c}$

$\mathrm{Defined\; as\; spacetime\; tensors\; representing\; the\; NP\; null\; vectors\; of\; the\; tetrad\; formalism}\left(\mathrm{see\; ?Physics,tetrads}\right),l_{\mathrm{\mu }},n_{\mathrm{\mu }},m_{\mathrm{\mu }},{\stackrel{\&conjugate0;}{m}}_{\mathrm{\mu }}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{IsTetrad}\,\mathrm{NullTetrad}\,\mathrm{OrthonormalTetrad}\,\mathrm{PetrovType}\,\mathrm{SegreType}\,\mathrm{TransformTetrad}\,\mathrm{WeylScalars}\,\mathrm{e\_}\,\mathrm{eta\_}\,\mathrm{gamma\_}\,\mathrm{l\_}\,\mathrm{lambda\_}\,\mathrm{m\_}\,\mathrm{mb\_}\,\mathrm{n\_}\right]$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{g\_}\left[\mathrm{sc}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

$\mathrm{Setting}\mathrm{lowercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{space}\mathrm{indices}$

$\mathrm{The\; Schwarzschild\; metric\; in\; coordinates}\left[r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]$

$\mathrm{Parameters:}\left[m\right]$

$\mathrm{Signature:}\left(\mathrm{-\; -\; -\; +}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\frac{r}{2m-r}& 0& 0& 0\\ 0& -r^2& 0& 0\\ 0& 0& -r^2{\sin \left(\mathrm{\theta }\right)}^2& 0\\ 0& 0& 0& \frac{r-2m}{r}\end{array}\right]$

$\mathrm{gamma\_}\left[\mathrm{definition}\right]$

${\mathrm{\gamma }}_{a,b,c}={▿}_{\mathrm{\nu }}\left({𝔢}_{a,\mathrm{\mu }}\right){𝔢}_{b\phantom{\mathrm{\mu }}}^{\phantom{b}\mathrm{\mu }}{𝔢}_{c\phantom{\mathrm{\nu }}}^{\phantom{c}\mathrm{\nu }}$

$\mathrm{lambda\_}\left[\mathrm{definition}\right]$

${\mathrm{\lambda }}_{a,b,c}=\left({▿}_{\mathrm{\nu }}\left({𝔢}_{a,\mathrm{\mu }}\right)-{▿}_{\mathrm{\mu }}\left({𝔢}_{a,\mathrm{\nu }}\right)\right){𝔢}_{b\phantom{\mathrm{\mu }}}^{\phantom{b}\mathrm{\mu }}{𝔢}_{c\phantom{\mathrm{\nu }}}^{\phantom{c}\mathrm{\nu }}$

$\mathrm{e\_}\left[b,\mathrm{\alpha }\right]\mathrm{e\_}\left[c,\mathrm{\beta }\right]$

${\mathrm{\gamma }}_{a,b,c}{𝔢}_{\phantom{}\phantom{b}\mathrm{\alpha }}^{\phantom{}b\phantom{\mathrm{\alpha }}}{𝔢}_{\phantom{}\phantom{c}\mathrm{\beta }}^{\phantom{}c\phantom{\mathrm{\beta }}}={▿}_{\mathrm{\nu }}\left({𝔢}_{a,\mathrm{\mu }}\right){𝔢}_{b,\mathrm{\alpha }}{𝔢}_{c,\mathrm{\beta }}{𝔢}_{\phantom{}\phantom{b,\mathrm{\mu }}}^{\phantom{}b,\mathrm{\mu }}{𝔢}_{\phantom{}\phantom{c,\mathrm{\nu }}}^{\phantom{}c,\mathrm{\nu }}$

$\mathrm{Simplify}\left(\mathrm{rhs}\left(\right)\right)=\mathrm{lhs}\left(\right)$

${▿}_{\mathrm{\beta }}\left({𝔢}_{a,\mathrm{\alpha }}\right)={\mathrm{\gamma }}_{a,b,c}{𝔢}_{\phantom{}\phantom{b}\mathrm{\alpha }}^{\phantom{}b\phantom{\mathrm{\alpha }}}{𝔢}_{\phantom{}\phantom{c}\mathrm{\beta }}^{\phantom{}c\phantom{\mathrm{\beta }}}$

$\mathrm{SubstituteTensor}\left(\,\right)$

${\mathrm{\lambda }}_{a,b,c}=\left(-{\mathrm{\gamma }}_{a,d,e}{𝔢}_{\phantom{}\phantom{d}\mathrm{\nu }}^{\phantom{}d\phantom{\mathrm{\nu }}}{𝔢}_{\phantom{}\phantom{e}\mathrm{\mu }}^{\phantom{}e\phantom{\mathrm{\mu }}}+{𝔢}_{\phantom{}\phantom{f}\mathrm{\mu }}^{\phantom{}f\phantom{\mathrm{\mu }}}{𝔢}_{\phantom{}\phantom{g}\mathrm{\nu }}^{\phantom{}g\phantom{\mathrm{\nu }}}{\mathrm{\gamma }}_{a,f,g}\right){𝔢}_{b\phantom{\mathrm{\mu }}}^{\phantom{b}\mathrm{\mu }}{𝔢}_{c\phantom{\mathrm{\nu }}}^{\phantom{c}\mathrm{\nu }}$

$\mathrm{Simplify}\left(\right)$

${\mathrm{\lambda }}_{a,b,c}=-{\mathrm{\gamma }}_{a,c,b}+{\mathrm{\gamma }}_{a,b,c}$

$+\mathrm{subs}\left(\left[a=b\,b=c\,c=a\right]\,\right)-\mathrm{subs}\left(\left[a=c\,b=a\,c=b\right]\,\right)$

${\mathrm{\lambda }}_{a,b,c}-{\mathrm{\lambda }}_{b,a,c}-{\mathrm{\lambda }}_{c,a,b}=2{\mathrm{\gamma }}_{a,b,c}$

$\mathrm{isolate}\left(\,\mathrm{gamma\_}\left[a,b,c\right]\right)$

${\mathrm{\gamma }}_{a,b,c}=\frac{{\mathrm{\lambda }}_{a,b,c}}{2}-\frac{{\mathrm{\lambda }}_{b,a,c}}{2}-\frac{{\mathrm{\lambda }}_{c,a,b}}{2}$

$\mathrm{gamma\_}\left[\mathrm{nonzero}\right]$

${\mathrm{\gamma }}_{a,b,c}=\left\{\left(1\,2\,2\right)=\frac{\mathrm{I}\sqrt{2m-r}}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\,\left(1\,3\,3\right)=\frac{\mathrm{I}\sqrt{2m-r}}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\,\left(1\,4\,4\right)=\frac{\mathrm{I}m}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{2m-r}}\,\left(2\,1\,2\right)=\frac{\mathrm{-I}\sqrt{2m-r}}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\,\left(2\,3\,3\right)=\frac{\cot \left(\mathrm{\theta }\right)}{r}\,\left(3\,1\,3\right)=\frac{\mathrm{-I}\sqrt{2m-r}}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\,\left(3\,2\,3\right)=-\frac{\cot \left(\mathrm{\theta }\right)}{r}\,\left(4\,1\,4\right)=\frac{\mathrm{-I}m}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{2m-r}}\right\}$

$\mathrm{lambda\_}\left[\mathrm{nonzero}\right]$

${\mathrm{\lambda }}_{a,b,c}=\left\{\left(2\,1\,2\right)=\frac{\mathrm{-I}\sqrt{2m-r}}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\,\left(2\,2\,1\right)=\frac{\mathrm{I}\sqrt{2m-r}}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\,\left(3\,1\,3\right)=\frac{\mathrm{-I}\sqrt{2m-r}}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\,\left(3\,2\,3\right)=-\frac{\cot \left(\mathrm{\theta }\right)}{r}\,\left(3\,3\,1\right)=\frac{\mathrm{I}\sqrt{2m-r}}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\,\left(3\,3\,2\right)=\frac{\cot \left(\mathrm{\theta }\right)}{r}\,\left(4\,1\,4\right)=\frac{\mathrm{-I}m}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{2m-r}}\,\left(4\,4\,1\right)=\frac{\mathrm{I}m}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{2m-r}}\right\}$

$\mathrm{gamma\_}\left[\mathrm{~1},1,1\right]$

$0$

$\mathrm{gamma\_}\left[\mathrm{~1},a,b,\mathrm{matrix}\right]$

${\mathrm{\gamma }}_{\phantom{}\phantom{1}a,b}^{\phantom{}1\phantom{a,b}}=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& \frac{\mathrm{-I}\sqrt{2m-r}}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}& 0& 0\\ 0& 0& \frac{\mathrm{-I}\sqrt{2m-r}}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}& 0\\ 0& 0& 0& \frac{\mathrm{-I}m}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{2m-r}}\end{array}\right]$

$\mathrm{Riemann}\left[a,b,c,d\right]=\mathrm{d\_}\left[d\right]\left(\mathrm{gamma\_}\left[a,b,c\right]\right)-\mathrm{d\_}\left[c\right]\left(\mathrm{gamma\_}\left[a,b,d\right]\right)+\mathrm{gamma\_}\left[a,b,f\right]\left(\mathrm{gamma\_}\left[f,c,d\right]-\mathrm{gamma\_}\left[f,d,c\right]\right)+\mathrm{gamma\_}\left[a,f,c\right]\mathrm{gamma\_}\left[f,b,d\right]-\mathrm{gamma\_}\left[a,f,d\right]\mathrm{gamma\_}\left[f,b,c\right]$

$R_{a,b,c,d}={\partial }_d\left({\mathrm{\gamma }}_{a,b,c}\right)-{\partial }_c\left({\mathrm{\gamma }}_{a,b,d}\right)+\left(-{\mathrm{\gamma }}_{c,f,d}+{\mathrm{\gamma }}_{d,f,c}\right){\mathrm{\gamma }}_{a,b\phantom{f}}^{\phantom{a}\phantom{,b}f}-{\mathrm{\gamma }}_{a,f,c}{\mathrm{\gamma }}_{b\phantom{f}d}^{\phantom{b}f\phantom{d}}+{\mathrm{\gamma }}_{a,f,d}{\mathrm{\gamma }}_{b\phantom{f}c}^{\phantom{b}f\phantom{c}}$

$\mathrm{ArrayElems}\left(\mathrm{TensorArray}\left(\left(\mathrm{lhs}-\mathrm{rhs}\right)\left(\right)\right)\right)$

$\varnothing$

${▿}_{\mathrm{\beta }}\left({𝔢}_{a,\mathrm{\alpha }}\right)={\mathrm{\gamma }}_{a,b,c}{𝔢}_{\phantom{}\phantom{b}\mathrm{\alpha }}^{\phantom{}b\phantom{\mathrm{\alpha }}}{𝔢}_{\phantom{}\phantom{c}\mathrm{\beta }}^{\phantom{}c\phantom{\mathrm{\beta }}}$

$\mathrm{ArrayElems}\left(\mathrm{TensorArray}\left(\left(\mathrm{lhs}-\mathrm{rhs}\right)\left(\right)\right)\right)$

$\varnothing$

Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.