check that a tensor is the Killing tensor for a given metric or connection - Maple Programming Help

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Tensor[CheckKillingTensor] - check that a tensor is the Killing tensor for a given metric or connection

Calling Sequences

CheckKillingTensor(g, T)

CheckKillingTensor(C, T)

Parameters

g    - a covariant metric tensor on a manifold M

T    - a symmetric covariant tensor on M, or a list of such

C    - an affine connection on a manifold M

Description

 • This program computes the symmetrized covariant derivative of the symmetric covariant tensor $T$ with respect to the Christoffel connection of the metric $g$ or the given connection $C$, that is, it computes the Killing tensor equation .
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form CheckKillingTensor(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:CheckKillingTensor.

Examples

 > with(DifferentialGeometry): with(Tensor):

Example 1.

Check that $\mathrm{K1}$ is a Killing tensor for the metric $g$.

 > DGsetup([x, y], M):
 M > g := evalDG((1/y)*dx &t dx + 1/x*dy &t dy);
 ${g}{:=}\frac{{1}}{{y}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}\frac{{1}}{{x}}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.1)
 M > K1 := evalDG((1/y^3)*dx &t dx &t dx - (1/x^3)*dy &t dy &t dy);
 ${\mathrm{K1}}{:=}\frac{{1}}{{{y}}^{{3}}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\frac{{1}}{{{x}}^{{3}}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.2)
 M > CheckKillingTensor(g, K1);
 ${0}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}$ (2.3)

Example 2.

Determine the equations for $A\left(y\right)$ and that must be satisfied for $\mathrm{K2}$ to be a Killing tensor for the metric $g$ from Example 1.

 M > K2 := evalDG(A(y)*dx &t dx &t dx + B(x)*dy &t dy &t dy);
 ${\mathrm{K2}}{:=}{A}{}\left({y}\right){}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{B}{}\left({x}\right){}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.4)
 M > P := CheckKillingTensor(g, K2);
 ${P}{:=}\frac{\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{A}{}\left({y}\right){}{y}{+}{3}{}{A}{}\left({y}\right)}{{4}{}{y}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{A}{}\left({y}\right){}{y}{+}{3}{}{A}{}\left({y}\right)}{{4}{}{y}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}\frac{{{x}}^{{3}}{}{B}{}\left({x}\right){+}{{y}}^{{3}}{}{A}{}\left({y}\right)}{{4}{}{{x}}^{{2}}{}{{y}}^{{2}}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}\frac{\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{A}{}\left({y}\right){}{y}{+}{3}{}{A}{}\left({y}\right)}{{4}{}{y}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\frac{{{x}}^{{3}}{}{B}{}\left({x}\right){+}{{y}}^{{3}}{}{A}{}\left({y}\right)}{{4}{}{{x}}^{{2}}{}{{y}}^{{2}}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}\frac{{{x}}^{{3}}{}{B}{}\left({x}\right){+}{{y}}^{{3}}{}{A}{}\left({y}\right)}{{4}{}{{x}}^{{2}}{}{{y}}^{{2}}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}\frac{{B}{\prime }{}\left({x}\right){}{x}{+}{3}{}{B}{}\left({x}\right)}{{4}{}{x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}\frac{\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{A}{}\left({y}\right){}{y}{+}{3}{}{A}{}\left({y}\right)}{{4}{}{y}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\frac{{{x}}^{{3}}{}{B}{}\left({x}\right){+}{{y}}^{{3}}{}{A}{}\left({y}\right)}{{4}{}{{x}}^{{2}}{}{{y}}^{{2}}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}\frac{{{x}}^{{3}}{}{B}{}\left({x}\right){+}{{y}}^{{3}}{}{A}{}\left({y}\right)}{{4}{}{{x}}^{{2}}{}{{y}}^{{2}}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}\frac{{B}{\prime }{}\left({x}\right){}{x}{+}{3}{}{B}{}\left({x}\right)}{{4}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{-}\frac{{{x}}^{{3}}{}{B}{}\left({x}\right){+}{{y}}^{{3}}{}{A}{}\left({y}\right)}{{4}{}{{x}}^{{2}}{}{{y}}^{{2}}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}\frac{{B}{\prime }{}\left({x}\right){}{x}{+}{3}{}{B}{}\left({x}\right)}{{4}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{B}{\prime }{}\left({x}\right){}{x}{+}{3}{}{B}{}\left({x}\right)}{{4}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}$ (2.5)
 M > Tools:-DGinfo(P, "CoefficientSet");
 $\left\{\frac{{1}}{{4}}{}\frac{\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{A}{}\left({y}\right)\right){}{y}{+}{3}{}{A}{}\left({y}\right)}{{y}}{,}\frac{{1}}{{4}}{}\frac{\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{B}{}\left({x}\right)\right){}{x}{+}{3}{}{B}{}\left({x}\right)}{{x}}{,}{-}\frac{{1}}{{4}}{}\frac{{{x}}^{{3}}{}{B}{}\left({x}\right){+}{{y}}^{{3}}{}{A}{}\left({y}\right)}{{{x}}^{{2}}{}{{y}}^{{2}}}\right\}$ (2.6)