Let $C$ be a connection on the tangent bundle of a manifold $M$ with a curvature tensor $C$. The Ricci tensor $R$ is the contraction of $C$$$ over the 1st and 3rd indices. In terms of index notation, $R_{\mathrm{ij}}\=R_{ \mathrm{ihj}}^h$ .

With the first calling sequence, the Ricci tensor for the Christoffel connection of the metric $g$ is computed. With the second calling sequence, the Ricci tensor is computed directly from the given curvature tensor.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form RicciTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-RicciTensor.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{C1}≔\mathrm{Connection}\left(x^2\left(\mathrm{D\_x}\&t\mathrm{dx}\right)\&t\mathrm{dy}-y^2\left(\mathrm{D\_x}\&t\mathrm{dy}\right)\&t\mathrm{dy}+yz\left(\mathrm{D\_x}\&t\mathrm{dz}\right)\&t\mathrm{dy}\right)$

$\mathrm{C1}:=x^2\mathrm{D\_x}\mathrm{dx}\mathrm{dy}-y^2\mathrm{D\_x}\mathrm{dy}\mathrm{dy}\+yz\mathrm{D\_x}\mathrm{dz}\mathrm{dy}$

$\mathrm{R1}≔\mathrm{CurvatureTensor}\left(\mathrm{C1}\right)$

$\mathrm{R1}:=2x\mathrm{D\_x}\mathrm{dx}\mathrm{dx}\mathrm{dy}-2x\mathrm{D\_x}\mathrm{dx}\mathrm{dy}\mathrm{dx}-y\mathrm{D\_x}\mathrm{dz}\mathrm{dy}\mathrm{dz}\+y\mathrm{D\_x}\mathrm{dz}\mathrm{dz}\mathrm{dy}$

$\mathrm{Ric1}≔\mathrm{RicciTensor}\left(\mathrm{R1}\right)$

$\mathrm{Ric1}:=2x\mathrm{dx}\mathrm{dy}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{FR}≔\mathrm{FrameData}\left(\left[\frac{x^2}{y}\mathrm{dx}\,\frac{z}{x}\mathrm{dy}\,xy\mathrm{dz}\right]\,\mathrm{M1}\right)\:$

$\mathrm{DGsetup}\left(\mathrm{FR}\right)$

$\mathrm{frame\; name:\; M1}$

$\mathrm{C2}≔\mathrm{Connection}\left(\left(\mathrm{E2}\&t\mathrm{\Θ1}\right)\&t\mathrm{\Θ2}\right)$

$\mathrm{C2}:=\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ2}$

$\mathrm{R2}≔\mathrm{CurvatureTensor}\left(\mathrm{C2}\right)$

$\mathrm{R2}:=-\frac{y\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ1}\mathrm{\Θ2}}{x^3}\+\frac{y\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ2}\mathrm{\Θ1}}{x^3}-\frac{\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ2}\mathrm{\Θ3}}{zxy}\+\frac{\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ3}\mathrm{\Θ2}}{zxy}$

$\mathrm{Ric2}≔\mathrm{RicciTensor}\left(\mathrm{R2}\right)$

$\mathrm{Ric2}:=\frac{y\mathrm{\Θ1}\mathrm{\Θ1}}{x^3}-\frac{\mathrm{\Θ1}\mathrm{\Θ3}}{zxy}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$g≔\mathrm{evalDG}\left(y\mathrm{dx}\&t\mathrm{dx}+z\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=\mathrm{dx}\mathrm{dx}y\+\mathrm{dy}\mathrm{dy}z\+\mathrm{dz}\mathrm{dz}$

$\mathrm{Ric3}≔\mathrm{RicciTensor}\left(g\right)$

$\mathrm{Ric3}:=\frac{1}{4}\frac{\mathrm{dx}\mathrm{dx}}{yz}\+\frac{1}{4}\frac{\left(y^2\+z\right)\mathrm{dy}\mathrm{dy}}{y^{2}z}\+\frac{1}{4}\frac{\mathrm{dy}\mathrm{dz}}{yz}\+\frac{1}{4}\frac{\mathrm{dz}\mathrm{dy}}{yz}\+\frac{1}{4}\frac{\mathrm{dz}\mathrm{dz}}{z^2}$