 calculate the Killing-Yano tensors for a given connection or a given metric - Maple Programming Help

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Tensor[KillingYanoTensors] - calculate the Killing-Yano tensors for a given connection or a given metric

Calling Sequences

KillingYanoTensors(gpoptions)

KillingYanoTensors(${\mathbf{Γ}}$, poptions)

Parameters

g         - a metric tensor on a manifold $M$

$\mathrm{Γ}$         - an affine connection on $M$

options   - any of the following keywords arguments: ansatz, unknowns, auxiliaryequations, coefficientvariables, parameters, output

Description

 • Let $▿$denote covariant differentiation with respect to the given connection $\mathrm{Γ}$, or with respect to the Christoffel connection defined by the metric $g$. A degree $p$ Killing-Yano tensor $T$ is a $p$-form, or equivalently, a skew-symmetric covariant tensor such that
 • The program KillingYanoTensors generates the defining system of 1st order PDE for a Killing-Yano tensor and uses pdsolve to find the solutions to these PDE.
 • The keyword argument coefficientvariables  allows the user to specify the coefficient functions in the Killing-Yano tensor $T$as functions of the variables  .
 • The exact form of the tensor $T$ can be specified with the keyword argument ansatz . For example, if the coordinates on the underlying manifold are and are defined $p$-forms, then one may solve for Killing-Yano tensors of the form . In this situation the unknown functions must be explicitly specified with the keyword argument unknowns, for example, unknowns
 • When using the keyword argument ansatz, additional algebraic or differential conditions may be imposed upon the unknowns using the keyword argument auxiliaryequations Here is a list of the auxiliary equations to be added to the Killing-Yano equations.
 • If the metric or connection $\mathrm{Γ}$ depends upon a number of unspecified parameters (either constants or functions), then the keyword argument parameterswhere is the list of parameters, will invoke case-splitting with respect to these parameters. Special values of the parameters, where either the number or the explicit form of the Killing-Yano tensors changes, are calculated.
 • With keyword argument output = the defining partial differential equations for the Killing-Yano tensors are returned. The option output = returns the general solution in terms of a number of arbitrary constants ${\mathrm{_C}}_{1}$, ${\mathrm{_C}}_{2}$, ... while the option output = returns a list of tensors which form a basis for the solution space. The default value of this keyword argument is output = $"list".$
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form KillingYanoTensors(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-KillingYanoTensors(...).

Examples

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

Example 1.

We find the Killing-Yano tensors of degree 2 and 3 for following metric $g.$

 > $\mathrm{DGsetup}\left(\left[u,v,x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)
 M > $g≔\mathrm{evalDG}\left(-\mathrm{exp}\left(2ax\right)\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}+\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}+\mathrm{dv}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}+\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)$
 ${g}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}{{ⅇ}}^{{2}{}{a}{}{x}}\right]{,}\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (2.2)

There are 2 Killing-Yano tensors of degree 2.

 M > $\mathrm{K2}≔\mathrm{KillingYanoTensors}\left(g,2\right)$
 ${\mathrm{K2}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.3)

There are 2 Killing-Yano tensors of degree 3.

 M > $\mathrm{K3}≔\mathrm{KillingYanoTensors}\left(g,3\right)$
 ${\mathrm{K3}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{3}\right]{,}\left[\left[\left[{1}{,}{3}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.4)

We can use the CovariantDerivative and SymmetrizeIndices commands to verify that the differential forms and satisfy the Killing-Yano equation. First we need the Christoffel connection for the metric $g.$

 M > $C≔\mathrm{Christoffel}\left(g\right)$
 ${C}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{2}{,}{1}{,}{3}\right]{,}{-}{a}{}{{ⅇ}}^{{2}{}{a}{}{x}}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{a}{}{{ⅇ}}^{{2}{}{a}{}{x}}\right]{,}\left[\left[{3}{,}{1}{,}{1}\right]{,}{a}{}{{ⅇ}}^{{2}{}{a}{}{x}}\right]\right]\right]\right)$ (2.5)

To check the Killing-Yano equation we take the covariant derivative of one of the tensors and symmetrize on the last two indices (the DifferentialGeometry convention is to place the index or slot for the covariant derivative last)

 M > $\mathrm{SymmetrizeIndices}\left(\mathrm{CovariantDerivative}\left(\mathrm{K2}\left[1\right],C\right),\left[2,3\right],"Symmetric"\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)$ (2.6)

 M > $\mathrm{SymmetrizeIndices}\left(\mathrm{CovariantDerivative}\left(\mathrm{K3}\left[1\right],C\right),\left[3,4\right],"Symmetric"\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)$ (2.7)

Example 2.

We can use the keyword arguments ansatz and unknowns to find a subset of the Killing-Yano tensors for this metric, say, the 2-forms, which are independent of and have coefficients which are functions of the variable alone.

 Sym > $T≔\mathrm{evalDG}\left(r\left(u\right)\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+s\left(u\right)\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}+t\left(u\right)\mathrm{dv}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)$
 ${T}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{s}{}\left({u}\right)\right]{,}\left[\left[{1}{,}{4}\right]{,}{r}{}\left({u}\right)\right]{,}\left[\left[{2}{,}{4}\right]{,}{t}{}\left({u}\right)\right]\right]\right]\right)$ (2.8)
 M > $\mathrm{vars}≔\left[r\left(u\right),s\left(u\right),t\left(u\right)\right]$
 ${\mathrm{vars}}{:=}\left[{r}{}\left({u}\right){,}{s}{}\left({u}\right){,}{t}{}\left({u}\right)\right]$ (2.9)
 M > $\mathrm{KillingYanoTensors}\left(g,2,\mathrm{ansatz}=T,\mathrm{unknowns}=\mathrm{vars}\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.10)

With the keyword argument output =the defining differential equations for the Killing-Yano tensors are returned.

 M > $\mathrm{KillingYanoTensors}\left(g,2,\mathrm{ansatz}=T,\mathrm{unknowns}=\mathrm{vars},\mathrm{output}="pde"\right)$
 $\left\{{0}{,}{a}{}{{ⅇ}}^{{2}{}{a}{}{x}}{}{s}{}\left({u}\right){,}{a}{}{{ⅇ}}^{{2}{}{a}{}{x}}{}{t}{}\left({u}\right){,}{-}\frac{{1}}{{2}}{}{a}{}{{ⅇ}}^{{2}{}{a}{}{x}}{}{s}{}\left({u}\right){,}{-}\frac{{1}}{{2}}{}{a}{}{{ⅇ}}^{{2}{}{a}{}{x}}{}{t}{}\left({u}\right){,}{-}\frac{{1}}{{2}}{}\frac{{ⅆ}}{{ⅆ}{u}}{}{r}{}\left({u}\right){,}{-}\frac{{1}}{{2}}{}\frac{{ⅆ}}{{ⅆ}{u}}{}{s}{}\left({u}\right){,}{-}\frac{{1}}{{2}}{}\frac{{ⅆ}}{{ⅆ}{u}}{}{t}{}\left({u}\right){,}\frac{{1}}{{2}}{}\frac{{ⅆ}}{{ⅆ}{u}}{}{t}{}\left({u}\right){,}\frac{{ⅆ}}{{ⅆ}{u}}{}{r}{}\left({u}\right){,}\frac{{ⅆ}}{{ⅆ}{u}}{}{s}{}\left({u}\right)\right\}$ (2.11)

Example 3.

Consider the following metric which depends upon an arbitrary function $f\left(y\right)$.

 M > $\mathrm{DifferentialGeometry}:-\mathrm{DGsetup}\left(\left[u,v,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.12)
 M > $g≔\mathrm{evalDG}\left(\mathrm{evalDG}\left(-\frac{\frac{3}{2}\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}}{{y}^{2}}-\frac{\frac{3}{2}\mathrm{dv}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}}{{y}^{2}}-3f\left(y\right)\mathrm{dv}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}+\frac{3\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}}{{y}^{2}}+\frac{3\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}}{{y}^{3}}\right)\right)$
 ${g}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}\frac{{3}}{{2}{}{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}\frac{{3}}{{2}{}{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}{3}{}{f}{}\left({y}\right)\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{3}}{{{y}}^{{2}}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{3}}{{{y}}^{{3}}}\right]\right]\right]\right)$ (2.13)

With the keyword argument parameters we can identity those special values of for which the metric admits a rank 2 Killing-Yano tensor.

 M > $\mathrm{KY}≔\mathrm{KillingYanoTensors}\left(g,2,\mathrm{parameters}=\left\{f\left(y\right)\right\}\right)$
 ${\mathrm{KY}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{3}}}\right]\right]\right]\right)\right]{,}\left[{}\right]{,}\left[\left\{{f}{}\left({y}\right){=}\frac{{\mathrm{_C1}}}{{{y}}^{{2}}}\right\}{,}\left\{{f}{}\left({y}\right){=}{f}{}\left({y}\right)\right\}\right]$ (2.14)

We see that there are no Killing-Yano tensors for generic choices of $f\left(y\right)$ and 1 Killing-Yano tensor when $f\left(y\right)=\frac{\mathrm{_C1}}{{y}^{2}}.$