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)\:$

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

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

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

$\mathrm{\_DG}\left(\left[\left[''connection''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,2\right]\,x^2\right]\,\left[\left[1\,2\,2\right]\,-y^2\right]\,\left[\left[1\,3\,2\right]\,yz\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''connection''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,2\right]\,x^2\right]\,\left[\left[1\,2\,2\right]\,-y^2\right]\,\left[\left[1\,3\,2\right]\,yz\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,2\right]\,2x\right]\,\left[\left[1\,1\,2\,1\right]\,-2x\right]\,\left[\left[1\,3\,2\,3\right]\,-y\right]\,\left[\left[1\,3\,3\,2\right]\,y\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,2\right]\,2x\right]\,\left[\left[1\,1\,2\,1\right]\,-2x\right]\,\left[\left[1\,3\,2\,3\right]\,-y\right]\,\left[\left[1\,3\,3\,2\right]\,y\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,2x\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,2x\right]\right]\right]\right)$

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

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

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

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

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

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

$\mathrm{\_DG}\left(\left[\left[''connection''\,\mathrm{M1}\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,1\,2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''connection''\,\mathrm{M1}\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,1\,2\right]\,1\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{M1}\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,1\,1\,2\right]\,-\frac{y}{x^3}\right]\,\left[\left[2\,1\,2\,1\right]\,\frac{y}{x^3}\right]\,\left[\left[2\,1\,2\,3\right]\,-\frac{1}{zxy}\right]\,\left[\left[2\,1\,3\,2\right]\,\frac{1}{zxy}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{M1}\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,1\,1\,2\right]\,-\frac{y}{x^3}\right]\,\left[\left[2\,1\,2\,1\right]\,\frac{y}{x^3}\right]\,\left[\left[2\,1\,2\,3\right]\,-\frac{1}{zxy}\right]\,\left[\left[2\,1\,3\,2\right]\,\frac{1}{zxy}\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{M1}\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,\frac{y}{x^3}\right]\,\left[\left[1\,3\right]\,-\frac{1}{zxy}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{M1}\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,\frac{y}{x^3}\right]\,\left[\left[1\,3\right]\,-\frac{1}{zxy}\right]\right]\right]\right)$

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

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

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

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,y\right]\,\left[\left[2\,2\right]\,z\right]\,\left[\left[3\,3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,y\right]\,\left[\left[2\,2\right]\,z\right]\,\left[\left[3\,3\right]\,1\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,\frac{1}{4yz}\right]\,\left[\left[2\,2\right]\,\frac{1}{4}\frac{y^2\+z}{y^{2}z}\right]\,\left[\left[2\,3\right]\,\frac{1}{4yz}\right]\,\left[\left[3\,2\right]\,\frac{1}{4yz}\right]\,\left[\left[3\,3\right]\,\frac{1}{4z^2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,\frac{1}{4yz}\right]\,\left[\left[2\,2\right]\,\frac{1}{4}\frac{y^2\+z}{y^{2}z}\right]\,\left[\left[2\,3\right]\,\frac{1}{4yz}\right]\,\left[\left[3\,2\right]\,\frac{1}{4yz}\right]\,\left[\left[3\,3\right]\,\frac{1}{4z^2}\right]\right]\right]\right)$