BachTensor - Maple Help

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 > $g≔\mathrm{evalDG}\left(-\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}+\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\mathrm{exp}\left(xy\right)\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}\right)$
 ${g}{:=}{{ⅇ}}^{{x}{}{y}}{}{\mathrm{du}}{}{\mathrm{du}}{-}\frac{{1}}{{2}}{}{\mathrm{du}}{}{\mathrm{dv}}{-}\frac{{1}}{{2}}{}{\mathrm{dv}}{}{\mathrm{du}}{+}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.2)
 M > $B≔\mathrm{BachTensor}\left(g\right)$
 ${B}{:=}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}{}{\mathrm{du}}{}{\mathrm{du}}$ (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 > $\mathrm{Gamma}≔\mathrm{Christoffel}\left(g\right)$
 ${\mathrm{Γ}}{:=}{-}{y}{}{{ⅇ}}^{{x}{}{y}}{}{\mathrm{D_v}}{}{\mathrm{du}}{}{\mathrm{dx}}{-}{x}{}{{ⅇ}}^{{x}{}{y}}{}{\mathrm{D_v}}{}{\mathrm{du}}{}{\mathrm{dy}}{-}{y}{}{{ⅇ}}^{{x}{}{y}}{}{\mathrm{D_v}}{}{\mathrm{dx}}{}{\mathrm{du}}{-}{x}{}{{ⅇ}}^{{x}{}{y}}{}{\mathrm{D_v}}{}{\mathrm{dy}}{}{\mathrm{du}}{-}\frac{{y}{}{{ⅇ}}^{{x}{}{y}}}{{2}}{}{\mathrm{D_x}}{}{\mathrm{du}}{}{\mathrm{du}}{-}\frac{{x}{}{{ⅇ}}^{{x}{}{y}}}{{2}}{}{\mathrm{D_y}}{}{\mathrm{du}}{}{\mathrm{du}}$ (2.5)
 M > $\mathrm{BachTensor}\left(g,\mathrm{Gamma}\right)$
 ${-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}{}{\mathrm{du}}{}{\mathrm{du}}$ (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 > $R≔\mathrm{CurvatureTensor}\left(g\right)$
 ${R}{:=}{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}{}{\mathrm{D_v}}{}{\mathrm{dx}}{}{\mathrm{du}}{}{\mathrm{dx}}{+}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right){}{\mathrm{D_v}}{}{\mathrm{dx}}{}{\mathrm{du}}{}{\mathrm{dy}}{-}{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}{}{\mathrm{D_v}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{du}}{-}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right){}{\mathrm{D_v}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{du}}{+}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right){}{\mathrm{D_v}}{}{\mathrm{dy}}{}{\mathrm{du}}{}{\mathrm{dx}}{+}{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}{}{\mathrm{D_v}}{}{\mathrm{dy}}{}{\mathrm{du}}{}{\mathrm{dy}}{-}{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right){}{\mathrm{D_v}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{du}}{-}{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}{}{\mathrm{D_v}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{du}}{+}\frac{{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}{}{\mathrm{D_x}}{}{\mathrm{du}}{}{\mathrm{du}}{}{\mathrm{dx}}{+}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}{}{\mathrm{D_x}}{}{\mathrm{du}}{}{\mathrm{du}}{}{\mathrm{dy}}{-}\frac{{{y}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}{}{\mathrm{D_x}}{}{\mathrm{du}}{}{\mathrm{dx}}{}{\mathrm{du}}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}{}{\mathrm{D_x}}{}{\mathrm{du}}{}{\mathrm{dy}}{}{\mathrm{du}}{+}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}{}{\mathrm{D_y}}{}{\mathrm{du}}{}{\mathrm{du}}{}{\mathrm{dx}}{+}\frac{{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}{}{\mathrm{D_y}}{}{\mathrm{du}}{}{\mathrm{du}}{}{\mathrm{dy}}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({x}{}{y}{+}{1}\right)}{{2}}{}{\mathrm{D_y}}{}{\mathrm{du}}{}{\mathrm{dx}}{}{\mathrm{du}}{-}\frac{{{x}}^{{2}}{}{{ⅇ}}^{{x}{}{y}}}{{2}}{}{\mathrm{D_y}}{}{\mathrm{du}}{}{\mathrm{dy}}{}{\mathrm{du}}$ (2.7)
 M > $C≔\mathrm{CottonTensor}\left(g\right)$
 ${C}{:=}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{2}}{}{y}{+}{{y}}^{{3}}{+}{2}{}{x}\right)}{{4}}{}{\mathrm{du}}{}{\mathrm{du}}{}{\mathrm{dx}}{+}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{3}}{+}{x}{}{{y}}^{{2}}{+}{2}{}{y}\right)}{{4}}{}{\mathrm{du}}{}{\mathrm{du}}{}{\mathrm{dy}}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{2}}{}{y}{+}{{y}}^{{3}}{+}{2}{}{x}\right)}{{4}}{}{\mathrm{du}}{}{\mathrm{dx}}{}{\mathrm{du}}{-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{3}}{+}{x}{}{{y}}^{{2}}{+}{2}{}{y}\right)}{{4}}{}{\mathrm{du}}{}{\mathrm{dy}}{}{\mathrm{du}}$ (2.8)
 M > $\mathrm{BachTensor}\left(g,\mathrm{Gamma},R,C\right)$
 ${-}\frac{{{ⅇ}}^{{x}{}{y}}{}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{+}{8}{}{x}{}{y}{+}{4}\right)}{{4}}{}{\mathrm{du}}{}{\mathrm{du}}$ (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 > $\mathrm{g0}≔\mathrm{evalDG}\left(-\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}+\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+xy\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}\right)$
 ${\mathrm{g0}}{:=}{x}{}{y}{}{\mathrm{du}}{}{\mathrm{du}}{-}\frac{{1}}{{2}}{}{\mathrm{du}}{}{\mathrm{dv}}{-}\frac{{1}}{{2}}{}{\mathrm{dv}}{}{\mathrm{du}}{+}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.11)
 M > $\mathrm{RicciTensor}\left(\mathrm{g0}\right)$
 ${0}{}{\mathrm{du}}{}{\mathrm{du}}$ (2.12)
 M > $\mathrm{g1}≔\mathrm{evalDG}\left(\mathrm{exp}\left(f\left(u,v,x,y\right)\right)\mathrm{g0}\right)$
 ${\mathrm{g1}}{:=}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}{}{x}{}{y}{}{\mathrm{du}}{}{\mathrm{du}}{-}\frac{{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}}{{2}}{}{\mathrm{du}}{}{\mathrm{dv}}{-}\frac{{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}}{{2}}{}{\mathrm{dv}}{}{\mathrm{du}}{+}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{{ⅇ}}^{{f}{}\left({u}{,}{v}{,}{x}{,}{y}\right)}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.13)
 M > $\mathrm{BachTensor}\left(\mathrm{g1}\right)$
 ${0}{}{\mathrm{du}}{}{\mathrm{du}}$ (2.14)