BachTensor - Maple Help
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Tensor[BachTensor] - calculate the Bach tensor of a metric

Calling Sequences

BachTensor(${\mathbit{g}}$)

BachTensor(${\mathbit{g}}$, ${\mathbf{Γ}}$)

BachTensor(${\mathbit{g}}$, G, R, C)

Parameters

g       - a metric tensor on the tangent bundle of a manifold

Γ       - (optional) the Christoffel connection of $g$

R       - (optional) the curvature tensor of $g$

C       - (optional) the Cotton tensor of $g$

Description

 • 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

he Bach tensor is trace-free: See A. Grover and P. Nurowski, J. Geom. Phys. 56, 450-484 (2006) for additional properties, applications and references.

 • 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.

Examples

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

Example 1.

Calculate the Bach tensor of a metric and check that it is trace-free.

 > $\mathrm{DGsetup}\left(\left[u,v,x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)
 M > $g≔\mathrm{evalDG}\left(-\mathrm{du}&s\mathrm{dv}+\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+{ⅇ}^{xy}\mathrm{du}&t\mathrm{du}\right)$
 ${g}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (2.2)
 M > $B≔\mathrm{BachTensor}\left(g\right)$
 ${B}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}\right]\right]\right]\right)$ (2.3)
 M > $\mathrm{TensorInnerProduct}\left(g,g,B\right)$
 ${0}$ (2.4)

Example 2.

Calculate the Bach tensor of a metric and Christoffel connection. We use the metric from the previous example.

 M > $\mathrm{Gamma}≔\mathrm{Christoffel}\left(g\right)$
 ${\mathrm{Γ}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{2}{,}{1}{,}{3}\right]{,}{-}{y}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{1}{,}{4}\right]{,}{-}{x}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{y}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{4}{,}{1}\right]{,}{-}{x}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{3}{,}{1}{,}{1}\right]{,}{-}\frac{{y}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{1}\right]{,}{-}\frac{{x}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\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[{2}{,}{1}{,}{3}\right]{,}{-}{y}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{1}{,}{4}\right]{,}{-}{x}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{y}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{4}{,}{1}\right]{,}{-}{x}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{3}{,}{1}{,}{1}\right]{,}{-}\frac{{y}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{1}\right]{,}{-}\frac{{x}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\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[{2}{,}{1}{,}{3}\right]{,}{-}{y}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{1}{,}{4}\right]{,}{-}{x}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{y}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{4}{,}{1}\right]{,}{-}{x}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{3}{,}{1}{,}{1}\right]{,}{-}\frac{{y}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{1}\right]{,}{-}\frac{{x}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\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[{2}{,}{1}{,}{3}\right]{,}{-}{y}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{1}{,}{4}\right]{,}{-}{x}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{y}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{4}{,}{1}\right]{,}{-}{x}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{3}{,}{1}{,}{1}\right]{,}{-}\frac{{y}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{1}\right]{,}{-}\frac{{x}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]\right]\right]\right)$ (2.5)
 M > $\mathrm{BachTensor}\left(g,\mathrm{Gamma}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}\right]\right]\right]\right)$ (2.6)

Example 3.

Calculate the Bach tensor of a metric Christoffel connection, curvature tensor and Cotton tensor. We use the metric and connection from the previous examples.

 M > $R≔\mathrm{CurvatureTensor}\left(g\right)$
 ${R}{:=}{\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[{2}{,}{3}{,}{1}{,}{3}\right]{,}{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{4}\right]{,}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{1}\right]{,}{-}{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{3}{,}{4}{,}{1}\right]{,}{-}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{4}{,}{1}{,}{3}\right]{,}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{4}{,}{1}{,}{4}\right]{,}{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{4}{,}{3}{,}{1}\right]{,}{-}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{4}{,}{4}{,}{1}\right]{,}{-}{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}\frac{{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{3}{,}{1}{,}{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{1}{,}{3}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{1}{,}{4}\right]{,}\frac{{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{4}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\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[{2}{,}{3}{,}{1}{,}{3}\right]{,}{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{4}\right]{,}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{1}\right]{,}{-}{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{3}{,}{4}{,}{1}\right]{,}{-}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{4}{,}{1}{,}{3}\right]{,}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{4}{,}{1}{,}{4}\right]{,}{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{4}{,}{3}{,}{1}\right]{,}{-}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{4}{,}{4}{,}{1}\right]{,}{-}{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}\frac{{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{3}{,}{1}{,}{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{1}{,}{3}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{1}{,}{4}\right]{,}\frac{{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{4}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\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[{2}{,}{3}{,}{1}{,}{3}\right]{,}{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{4}\right]{,}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{1}\right]{,}{-}{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{3}{,}{4}{,}{1}\right]{,}{-}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{4}{,}{1}{,}{3}\right]{,}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{4}{,}{1}{,}{4}\right]{,}{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{4}{,}{3}{,}{1}\right]{,}{-}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{4}{,}{4}{,}{1}\right]{,}{-}{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}\frac{{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{3}{,}{1}{,}{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{1}{,}{3}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{1}{,}{4}\right]{,}\frac{{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{4}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\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[{2}{,}{3}{,}{1}{,}{3}\right]{,}{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{4}\right]{,}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{1}\right]{,}{-}{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{3}{,}{4}{,}{1}\right]{,}{-}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{4}{,}{1}{,}{3}\right]{,}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{4}{,}{1}{,}{4}\right]{,}{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{2}{,}{4}{,}{3}{,}{1}\right]{,}{-}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)\right]{,}\left[\left[{2}{,}{4}{,}{4}{,}{1}\right]{,}{-}{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}\frac{{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{3}{,}{1}{,}{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{1}{,}{3}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{1}{,}{4}\right]{,}\frac{{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}\right]{,}\left[\left[{4}{,}{1}{,}{4}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}\right]\right]\right]\right)$ (2.7)
 M > $C≔\mathrm{CottonTensor}\left(g\right)$
 ${C}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{3}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{2}}{}{y}{+}{{y}}^{{3}}{+}{2}{}{x}\right)}{{4}}\right]{,}\left[\left[{1}{,}{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{3}}{+}{x}{}{{y}}^{{2}}{+}{2}{}{y}\right)}{{4}}\right]{,}\left[\left[{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{2}}{}{y}{+}{{y}}^{{3}}{+}{2}{}{x}\right)}{{4}}\right]{,}\left[\left[{1}{,}{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{3}}{+}{x}{}{{y}}^{{2}}{+}{2}{}{y}\right)}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{3}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{2}}{}{y}{+}{{y}}^{{3}}{+}{2}{}{x}\right)}{{4}}\right]{,}\left[\left[{1}{,}{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{3}}{+}{x}{}{{y}}^{{2}}{+}{2}{}{y}\right)}{{4}}\right]{,}\left[\left[{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{2}}{}{y}{+}{{y}}^{{3}}{+}{2}{}{x}\right)}{{4}}\right]{,}\left[\left[{1}{,}{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{3}}{+}{x}{}{{y}}^{{2}}{+}{2}{}{y}\right)}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{3}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{2}}{}{y}{+}{{y}}^{{3}}{+}{2}{}{x}\right)}{{4}}\right]{,}\left[\left[{1}{,}{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{3}}{+}{x}{}{{y}}^{{2}}{+}{2}{}{y}\right)}{{4}}\right]{,}\left[\left[{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{2}}{}{y}{+}{{y}}^{{3}}{+}{2}{}{x}\right)}{{4}}\right]{,}\left[\left[{1}{,}{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{3}}{+}{x}{}{{y}}^{{2}}{+}{2}{}{y}\right)}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{3}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{2}}{}{y}{+}{{y}}^{{3}}{+}{2}{}{x}\right)}{{4}}\right]{,}\left[\left[{1}{,}{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{3}}{+}{x}{}{{y}}^{{2}}{+}{2}{}{y}\right)}{{4}}\right]{,}\left[\left[{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{2}}{}{y}{+}{{y}}^{{3}}{+}{2}{}{x}\right)}{{4}}\right]{,}\left[\left[{1}{,}{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{3}}{+}{x}{}{{y}}^{{2}}{+}{2}{}{y}\right)}{{4}}\right]\right]\right]\right)$ (2.8)
 M > $\mathrm{BachTensor}\left(g,\mathrm{Gamma},R,C\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}\right]\right]\right]\right)$ (2.9)

Example 3.

In four dimensions, the Bach tensor is an obstruction to a metric being conformal to an Einstein metric. Here we check that the Bach tensor vanishes on a metric conformal to a Ricci-flat metric in four dimensions.

 M > $\mathrm{DGsetup}\left(\left[u,v,x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.10)
 M > $\mathrm{g0}≔\mathrm{evalDG}\left(-\mathrm{du}&s\mathrm{dv}+\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+xy\mathrm{du}&t\mathrm{du}\right)$
 ${\mathrm{g0}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{x}{}{y}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{x}{}{y}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{x}{}{y}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{x}{}{y}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (2.11)
 M > $\mathrm{RicciTensor}\left(\mathrm{g0}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)$ (2.12)
 M > $\mathrm{g1}≔\mathrm{evalDG}\left({ⅇ}^{f\left(u,v,x,y\right)}\mathrm{g0}\right)$
 ${\mathrm{g1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}{}{x}{}{y}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-}\frac{{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}\right]{,}\left[\left[{4}{,}{4}\right]{,}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}\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]{,}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}{}{x}{}{y}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-}\frac{{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}\right]{,}\left[\left[{4}{,}{4}\right]{,}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}\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]{,}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}{}{x}{}{y}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-}\frac{{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}\right]{,}\left[\left[{4}{,}{4}\right]{,}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}\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]{,}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}{}{x}{}{y}\right]{,}\left[\left[{1}{,}{2}\right]{,}{-}\frac{{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}\right]{,}\left[\left[{4}{,}{4}\right]{,}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}\right]\right]\right]\right)$ (2.13)
 M > $\mathrm{BachTensor}\left(\mathrm{g1}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)$ (2.14)

 See Also