 KillingBracket - Maple Help

Tensor[KillingBracket] - a covariant form of the Schouten bracket for symmetric tensors

Calling Sequences

KillingBracket(g, R, S)

Parameters

g         - a covariant metric tensor on a manifold $M$

R, S      - symmetric covariant tensor fields on $M$ Description

 • If $R$ and $S$ are symmetric covariant tensor fields of rank $r$ and $s$, respectively, then T = KillingBracket(g, R, S) is a symmetric covariant tensor field of rank $r+s-1$. If $R$ and $S$ correspond to Killing tensors for the metric $g$, then $T$ is also a Killing tensor.
 • KillingBracket(g, R, S) can be defined in terms of the Schouten bracket for symmetric contravariant tensors by using the metric $g$ to raise all the indices on the tensors $R$ and $S$ to obtain contravariant tensors $U$ and $V$, then computing the Schouten bracket W = TensorBrackets(g, U, V, "Schouten") and then lowering all the indices of $W$. An explicit formula for the KillingBracket can be found in the article of N. M. J. Woodhouse.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form KillingBracket(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-KillingBracket. Examples

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

Example 1.

We compute the Killing bracket of two rank-2 tensors.

$R$ and $S$ are Killing tensors on $M$ with respect to $g$.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$
 > $g≔\mathrm{evalDG}\left(x\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+x\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
 ${g}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{x}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{x}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{x}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{x}}{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{x}\right]\right]\right]\right)$ (2.1)
 M > $R≔\mathrm{evalDG}\left(z{x}^{2}\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}-\frac{1}{2}{x}^{3}\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}-\frac{1}{2}{x}^{3}\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)$
 ${R}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{z}{}{{x}}^{{2}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}\frac{{{x}}^{{3}}}{{2}}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}\frac{{{x}}^{{3}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{z}{}{{x}}^{{2}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}\frac{{{x}}^{{3}}}{{2}}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}\frac{{{x}}^{{3}}}{{2}}\right]\right]\right]\right)$ (2.2)
 M > $S≔\mathrm{evalDG}\left(\frac{1}{2}{x}^{2}\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}+\frac{1}{2}{x}^{2}\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)$
 ${S}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}\frac{{{x}}^{{2}}}{{2}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{{{x}}^{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}\frac{{{x}}^{{2}}}{{2}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{{{x}}^{{2}}}{{2}}\right]\right]\right]\right)$ (2.3)
 M > $\mathrm{CheckKillingTensor}\left(g,\left[R,S\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)\right]$ (2.4)

We compute the Killing bracket of $R$ and $S$ and verify that the result is a Killing tensor.

 M > $T≔\mathrm{KillingBracket}\left(g,R,S\right)$
 ${T}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{-}{{x}}^{{3}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{-}{{x}}^{{3}}\right]\right]\right]\right)$ (2.5)
 M > $\mathrm{CheckKillingTensor}\left(g,T\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)$ (2.6)
 M >