Let $C$ be a connection on the tangent bundle of a manifold $M$ of dimension$m\>1.$ Let the curvature tensor of $C$ be $R$ and the Ricci tensor of $C$ be $Q$. The Weyl projective curvature of $C$ is the tensor $P$ of type $\left(\genfrac{}{}{0ex}{}{1}{3}\right)$ given by

With the first calling sequence the projective curvature tensor of the Christoffel connection of the metric g is computed. With the second calling sequence, the projective curvature tensor is computed directly from the given connection.  With the third calling sequence, the projective curvature 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 ProjectiveCurvature(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-ProjectiveCurvature.

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

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

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

$g≔\mathrm{evalDG}\left(\exp \left(\mathrm{\lambda }x\right)\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)\right)$

$g:={\ⅇ}^{\mathrm{\λ}x}\mathrm{dx}\mathrm{dx}\+{\ⅇ}^{\mathrm{\λ}x}\mathrm{dy}\mathrm{dy}\+{\ⅇ}^{\mathrm{\λ}x}\mathrm{dz}\mathrm{dz}$

$\mathrm{ProjectiveCurvatureTensor}\left(g\right)$

$\frac{1}{8}{\mathrm{\λ}}^2\mathrm{D\_x}\mathrm{dy}\mathrm{dx}\mathrm{dy}-\frac{1}{8}{\mathrm{\λ}}^2\mathrm{D\_x}\mathrm{dy}\mathrm{dy}\mathrm{dx}\+\frac{1}{8}{\mathrm{\λ}}^2\mathrm{D\_x}\mathrm{dz}\mathrm{dx}\mathrm{dz}-\frac{1}{8}{\mathrm{\λ}}^2\mathrm{D\_x}\mathrm{dz}\mathrm{dz}\mathrm{dx}-\frac{1}{8}{\mathrm{\λ}}^2\mathrm{D\_y}\mathrm{dz}\mathrm{dy}\mathrm{dz}\+\frac{1}{8}{\mathrm{\λ}}^2\mathrm{D\_y}\mathrm{dz}\mathrm{dz}\mathrm{dy}\+\frac{1}{8}{\mathrm{\λ}}^2\mathrm{D\_z}\mathrm{dy}\mathrm{dy}\mathrm{dz}-\frac{1}{8}{\mathrm{\λ}}^2\mathrm{D\_z}\mathrm{dy}\mathrm{dz}\mathrm{dy}$

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

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

$C≔\mathrm{Connection}\left(w\left(y\right)\left(\mathrm{D\_x}\&t\mathrm{dx}\right)\&t\mathrm{dx}+f\left(u\right)\mathrm{D\_y}\&t\left(\mathrm{dx}\&s\mathrm{dz}\right)\right)$

$C:=w\left(y\right)\mathrm{D\_x}\mathrm{dx}\mathrm{dx}\+\frac{1}{2}f\left(u\right)\mathrm{D\_y}\mathrm{dx}\mathrm{dz}\+\frac{1}{2}f\left(u\right)\mathrm{D\_y}\mathrm{dz}\mathrm{dx}$

$\mathrm{ProjectiveCurvatureTensor}\left(C\right)$

$-\frac{8}{15}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_x}\mathrm{dx}\mathrm{dx}\mathrm{dy}\+\frac{8}{15}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_x}\mathrm{dx}\mathrm{dy}\mathrm{dx}-\frac{1}{2}f\left(u\right)w\left(y\right)\mathrm{D\_y}\mathrm{dx}\mathrm{dx}\mathrm{dz}\+\frac{1}{2}f\left(u\right)w\left(y\right)\mathrm{D\_y}\mathrm{dx}\mathrm{dz}\mathrm{dx}-\frac{1}{2}\left(\frac{\ⅆ}{\ⅆu}f\left(u\right)\right)\mathrm{D\_y}\mathrm{dx}\mathrm{dz}\mathrm{du}\+\frac{1}{2}\left(\frac{\ⅆ}{\ⅆu}f\left(u\right)\right)\mathrm{D\_y}\mathrm{dx}\mathrm{du}\mathrm{dz}\+\frac{2}{15}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_y}\mathrm{dy}\mathrm{dx}\mathrm{dy}-\frac{2}{15}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_y}\mathrm{dy}\mathrm{dy}\mathrm{dx}-\frac{1}{2}\left(\frac{\ⅆ}{\ⅆu}f\left(u\right)\right)\mathrm{D\_y}\mathrm{dz}\mathrm{dx}\mathrm{du}\+\frac{1}{2}\left(\frac{\ⅆ}{\ⅆu}f\left(u\right)\right)\mathrm{D\_y}\mathrm{dz}\mathrm{du}\mathrm{dx}-\frac{4}{15}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_z}\mathrm{dx}\mathrm{dy}\mathrm{dz}\+\frac{4}{15}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_z}\mathrm{dx}\mathrm{dz}\mathrm{dy}-\frac{1}{15}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_z}\mathrm{dy}\mathrm{dx}\mathrm{dz}\+\frac{1}{15}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_z}\mathrm{dy}\mathrm{dz}\mathrm{dx}\+\frac{1}{5}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_z}\mathrm{dz}\mathrm{dx}\mathrm{dy}-\frac{1}{5}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_z}\mathrm{dz}\mathrm{dy}\mathrm{dx}-\frac{4}{15}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_u}\mathrm{dx}\mathrm{dy}\mathrm{du}\+\frac{4}{15}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_u}\mathrm{dx}\mathrm{du}\mathrm{dy}-\frac{1}{15}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_u}\mathrm{dy}\mathrm{dx}\mathrm{du}\+\frac{1}{15}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_u}\mathrm{dy}\mathrm{du}\mathrm{dx}\+\frac{1}{5}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_u}\mathrm{du}\mathrm{dx}\mathrm{dy}-\frac{1}{5}\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)\mathrm{D\_u}\mathrm{du}\mathrm{dy}\mathrm{dx}$

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

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

$g≔\mathrm{evalDG}\left(-\mathrm{dt}\&t\mathrm{dt}+\exp \left(a\left(t\right)\right)\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)\right)$

$g:=-\mathrm{dt}\mathrm{dt}\+{\ⅇ}^{a\left(t\right)}\mathrm{dx}\mathrm{dx}\+{\ⅇ}^{a\left(t\right)}\mathrm{dy}\mathrm{dy}\+{\ⅇ}^{a\left(t\right)}\mathrm{dz}\mathrm{dz}$

$R≔\mathrm{CurvatureTensor}\left(g\right)\:$

$\mathrm{ProjectiveCurvatureTensor}\left(R\right)$

$\frac{1}{3}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_t}\mathrm{dx}\mathrm{dt}\mathrm{dx}-\frac{1}{3}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_t}\mathrm{dx}\mathrm{dx}\mathrm{dt}\+\frac{1}{3}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_t}\mathrm{dy}\mathrm{dt}\mathrm{dy}-\frac{1}{3}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_t}\mathrm{dy}\mathrm{dy}\mathrm{dt}\+\frac{1}{3}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_t}\mathrm{dz}\mathrm{dt}\mathrm{dz}-\frac{1}{3}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_t}\mathrm{dz}\mathrm{dz}\mathrm{dt}-\frac{1}{6}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_x}\mathrm{dy}\mathrm{dx}\mathrm{dy}\+\frac{1}{6}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_x}\mathrm{dy}\mathrm{dy}\mathrm{dx}-\frac{1}{6}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_x}\mathrm{dz}\mathrm{dx}\mathrm{dz}\+\frac{1}{6}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_x}\mathrm{dz}\mathrm{dz}\mathrm{dx}\+\frac{1}{6}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_y}\mathrm{dx}\mathrm{dx}\mathrm{dy}-\frac{1}{6}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_y}\mathrm{dx}\mathrm{dy}\mathrm{dx}-\frac{1}{6}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_y}\mathrm{dz}\mathrm{dy}\mathrm{dz}\+\frac{1}{6}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_y}\mathrm{dz}\mathrm{dz}\mathrm{dy}\+\frac{1}{6}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_z}\mathrm{dx}\mathrm{dx}\mathrm{dz}-\frac{1}{6}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_z}\mathrm{dx}\mathrm{dz}\mathrm{dx}\+\frac{1}{6}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_z}\mathrm{dy}\mathrm{dy}\mathrm{dz}-\frac{1}{6}\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}a\left(t\right)\right)\right){\ⅇ}^{a\left(t\right)}\mathrm{D\_z}\mathrm{dy}\mathrm{dz}\mathrm{dy}$