Let $g_{\mathrm{ab}}$ be a metric (of any signature) on the tangent bundle of a manifold $M$ of dimension$$$n\>2.$ The metric determines: the covariant derivative ${\nabla }_a$, the Schouten tensor $P_{\mathrm{ab}}$, the Weyl tensor $W_{\mathrm{abcd}}$ and the Cotton tensor $C_{\mathrm{abc}}\.$ The Bach tensor is defined as

The first calling sequence computes $B_{\mathrm{ab}}$ directly from the given metric using the formula above. The second calling sequence computes $B_{\mathrm{ab}}$ from the given metric and Christoffel connection. The third calling sequence computes $B_{\mathrm{ab}}$ directly from the given metric Christoffel connection, curvature and Cotton tensors.

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

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

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

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

$g≔\mathrm{evalDG}\left(-\mathrm{du}\&s\mathrm{dv}+\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\exp \left(xy\right)\mathrm{du}\&t\mathrm{du}\right)$

$g:={\ⅇ}^{xy}\mathrm{du}\mathrm{du}-\frac{1}{2}\mathrm{du}\mathrm{dv}-\frac{1}{2}\mathrm{dv}\mathrm{du}+\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dy}$

$B≔\mathrm{BachTensor}\left(g\right)$

$B:=-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\mathrm{du}\mathrm{du}$

$\mathrm{TensorInnerProduct}\left(g\,g\,B\right)$

$0$

$\mathrm{Gamma}≔\mathrm{Christoffel}\left(g\right)$

$\mathrm{\Γ}:=-y{\ⅇ}^{xy}\mathrm{D\_v}\mathrm{du}\mathrm{dx}-x{\ⅇ}^{xy}\mathrm{D\_v}\mathrm{du}\mathrm{dy}-y{\ⅇ}^{xy}\mathrm{D\_v}\mathrm{dx}\mathrm{du}-x{\ⅇ}^{xy}\mathrm{D\_v}\mathrm{dy}\mathrm{du}-\frac{y{\ⅇ}^{xy}}{2}\mathrm{D\_x}\mathrm{du}\mathrm{du}-\frac{x{\ⅇ}^{xy}}{2}\mathrm{D\_y}\mathrm{du}\mathrm{du}$

$\mathrm{BachTensor}\left(g\,\mathrm{Gamma}\right)$

$-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\mathrm{du}\mathrm{du}$

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

$R:=y^2{\ⅇ}^{xy}\mathrm{D\_v}\mathrm{dx}\mathrm{du}\mathrm{dx}+{\ⅇ}^{xy}\left(xy+1\right)\mathrm{D\_v}\mathrm{dx}\mathrm{du}\mathrm{dy}-y^2{\ⅇ}^{xy}\mathrm{D\_v}\mathrm{dx}\mathrm{dx}\mathrm{du}-{\ⅇ}^{xy}\left(xy+1\right)\mathrm{D\_v}\mathrm{dx}\mathrm{dy}\mathrm{du}+{\ⅇ}^{xy}\left(xy+1\right)\mathrm{D\_v}\mathrm{dy}\mathrm{du}\mathrm{dx}+x^2{\ⅇ}^{xy}\mathrm{D\_v}\mathrm{dy}\mathrm{du}\mathrm{dy}-{\ⅇ}^{xy}\left(xy+1\right)\mathrm{D\_v}\mathrm{dy}\mathrm{dx}\mathrm{du}-x^2{\ⅇ}^{xy}\mathrm{D\_v}\mathrm{dy}\mathrm{dy}\mathrm{du}+\frac{y^2{\ⅇ}^{xy}}{2}\mathrm{D\_x}\mathrm{du}\mathrm{du}\mathrm{dx}+\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\mathrm{D\_x}\mathrm{du}\mathrm{du}\mathrm{dy}-\frac{y^2{\ⅇ}^{xy}}{2}\mathrm{D\_x}\mathrm{du}\mathrm{dx}\mathrm{du}-\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\mathrm{D\_x}\mathrm{du}\mathrm{dy}\mathrm{du}+\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\mathrm{D\_y}\mathrm{du}\mathrm{du}\mathrm{dx}+\frac{x^2{\ⅇ}^{xy}}{2}\mathrm{D\_y}\mathrm{du}\mathrm{du}\mathrm{dy}-\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\mathrm{D\_y}\mathrm{du}\mathrm{dx}\mathrm{du}-\frac{x^2{\ⅇ}^{xy}}{2}\mathrm{D\_y}\mathrm{du}\mathrm{dy}\mathrm{du}$

$C≔\mathrm{CottonTensor}\left(g\right)$

$C:=\frac{{\ⅇ}^{xy}\left(x^{2}y+y^3+2x\right)}{4}\mathrm{du}\mathrm{du}\mathrm{dx}+\frac{{\ⅇ}^{xy}\left(x^3+x{y}^2+2y\right)}{4}\mathrm{du}\mathrm{du}\mathrm{dy}-\frac{{\ⅇ}^{xy}\left(x^{2}y+y^3+2x\right)}{4}\mathrm{du}\mathrm{dx}\mathrm{du}-\frac{{\ⅇ}^{xy}\left(x^3+x{y}^2+2y\right)}{4}\mathrm{du}\mathrm{dy}\mathrm{du}$

$\mathrm{BachTensor}\left(g\,\mathrm{Gamma}\,R\,C\right)$

$-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\mathrm{du}\mathrm{du}$

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

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

$\mathrm{g0}≔\mathrm{evalDG}\left(-\mathrm{du}\&s\mathrm{dv}+\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+xy\mathrm{du}\&t\mathrm{du}\right)$

$\mathrm{g0}:=xy\mathrm{du}\mathrm{du}-\frac{1}{2}\mathrm{du}\mathrm{dv}-\frac{1}{2}\mathrm{dv}\mathrm{du}+\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dy}$

$\mathrm{RicciTensor}\left(\mathrm{g0}\right)$

$0\mathrm{du}\mathrm{du}$

$\mathrm{g1}≔\mathrm{evalDG}\left(\exp \left(f\left(u\,v\,x\,y\right)\right)\mathrm{g0}\right)$

$\mathrm{g1}:={\ⅇ}^{f\left(u\,v\,x\,y\right)}xy\mathrm{du}\mathrm{du}-\frac{{\ⅇ}^{f\left(u\,v\,x\,y\right)}}{2}\mathrm{du}\mathrm{dv}-\frac{{\ⅇ}^{f\left(u\,v\,x\,y\right)}}{2}\mathrm{dv}\mathrm{du}+{\ⅇ}^{f\left(u\,v\,x\,y\right)}\mathrm{dx}\mathrm{dx}+{\ⅇ}^{f\left(u\,v\,x\,y\right)}\mathrm{dy}\mathrm{dy}$

$\mathrm{BachTensor}\left(\mathrm{g1}\right)$

$0\mathrm{du}\mathrm{du}$