 CottonTensor - Maple Help

Tensor[CottonTensor] - calculate the Cotton tensor for a metric

Calling Sequences

CottonTensor(g, C, R)

Parameters

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

C        - (optional) the Christoffel connection for the metric g

R        - (optional) the curvature tensor of the metric g
keyword  -  (optional) the keyword argument indextype = ["cov_bas", "cov_bas", "cov_bas"] or indextype = ["con_bas", "con_bas"] Description

Let ${P}_{\mathrm{ab}}$ be the Schouten tensor for the metric ${g}_{\mathrm{ab}}$ with covariant derivative ${\nabla }_{a}.$  The Cotton tensor is defined (in any dimension) by

In 3-dimensions an alternative form of the Cotton tensor is a symmetric tensor density of weight one:

Here ${e}^{\mathrm{iab}}$ denotes the contravariant permutation symbol, which is a tensor density of weight one.  If desired one can convert ${Y}^{\mathrm{ih}}$ to a tensor by multiplication with a suitable MetricDensity.

 • ${C}_{\mathrm{abc}}$ is completely trace-free and anti-symmetric on its last two indices; it is divergence-free on its first index. The tensor ${Y}^{\mathrm{ij}}$ is symmetric, trace-free, and divergence-free. ${C}_{\mathrm{abc}}$ is conformally invariant. ${Y}^{\mathrm{ij}}$ is a relative conformal invariant.
 • If the optional arguments are not supplied, the Christoffel symbol and curvature tensor are computed directly from the metric, otherwise the supplied Christoffel symbol and curvature tensor are used.
 • The default output is ${C}_{\mathrm{abc}}$ as shown above. This output can also be obtained with keyword argument indextype = ["cov_bas", "cov_bas", "cov_bas"]. The keyword argument indextype = ["con_bas", "con_bas"] returns the tensor ${Y}^{\mathrm{ij}}$ described above.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form CottonTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-CottonTensor. Examples

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

Example 1.

First create a 3 dimensional manifold M and define a metric g1 on the tangent space of M.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)
 > $\mathrm{g1}≔\mathrm{evalDG}\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+xz\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)+{y}^{2}\left(\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}+\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)\right)$
 ${\mathrm{g1}}{:=}{\mathrm{dx}}{}{\mathrm{dx}}{+}{x}{}{z}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{x}{}{z}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{{y}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}{{y}}^{{2}}{}{\mathrm{dz}}{}{\mathrm{dy}}$ (2.2)
 > $\mathrm{CotTen1}≔\mathrm{CottonTensor}\left(\mathrm{g1}\right)$
 ${\mathrm{CotTen1}}{:=}{-}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{5}}}{+}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{5}}}{+}\frac{{x}{}\left({2}{}{z}{}{y}{+}{5}\right){}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{4}}}{-}\frac{{x}{}\left({2}{}{z}{}{y}{+}{5}\right){}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{4}}}{-}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dz}}}{{{y}}^{{3}}}{+}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dy}}}{{{y}}^{{3}}}$ (2.3)

Check that the Cotton tensor CotTen1 is trace-free.

 > $\mathrm{h1}≔\mathrm{InverseMetric}\left(\mathrm{g1}\right)$
 ${\mathrm{h1}}{:=}{\mathrm{D_x}}{}{\mathrm{D_x}}{-}\frac{{x}{}{z}{}{\mathrm{D_x}}{}{\mathrm{D_z}}}{{{y}}^{{2}}}{+}\frac{{\mathrm{D_y}}{}{\mathrm{D_z}}}{{{y}}^{{2}}}{-}\frac{{x}{}{z}{}{\mathrm{D_z}}{}{\mathrm{D_x}}}{{{y}}^{{2}}}{+}\frac{{\mathrm{D_z}}{}{\mathrm{D_y}}}{{{y}}^{{2}}}{+}\frac{{{x}}^{{2}}{}{{z}}^{{2}}{}{\mathrm{D_z}}{}{\mathrm{D_z}}}{{{y}}^{{4}}}$ (2.4)
 > $\mathrm{ContractIndices}\left(\mathrm{h1},\mathrm{CotTen1},\left[\left[1,1\right],\left[2,2\right]\right]\right)$
 ${0}{}{\mathrm{dx}}$ (2.5)

Check that the Cotton tensor is divergence-free on its first index.

 > $\mathrm{Ch1}≔\mathrm{Christoffel}\left(\mathrm{g1}\right)$
 ${\mathrm{Ch1}}{:=}\frac{{1}}{{2}}{}\frac{{{x}}^{{2}}{}{z}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{2}}}{+}\frac{{1}}{{2}}{}\frac{{{x}}^{{2}}{}{z}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{2}}}{-}\frac{{2}{}{x}{}{z}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{y}}{+}\frac{{1}}{{2}}{}{x}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{x}{}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dy}}{-}\frac{{1}}{{2}}{}\frac{{x}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{{x}{}{\mathrm{D_y}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{2}}}{+}\frac{{2}{}{\mathrm{D_y}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{y}}{+}\frac{{z}{}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dx}}}{{{y}}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{{{x}}^{{3}}{}{{z}}^{{2}}{}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{4}}}{+}\frac{{1}}{{2}}{}\frac{{x}{}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dz}}}{{{y}}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{{{x}}^{{3}}{}{{z}}^{{2}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{4}}}{+}\frac{{2}{}{{x}}^{{2}}{}{{z}}^{{2}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{{y}}^{{3}}}{-}\frac{{1}}{{2}}{}\frac{{{x}}^{{2}}{}{z}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dz}}}{{{y}}^{{2}}}{+}\frac{{1}}{{2}}{}\frac{{x}{}{\mathrm{D_z}}{}{\mathrm{dz}}{}{\mathrm{dx}}}{{{y}}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{{{x}}^{{2}}{}{z}{}{\mathrm{D_z}}{}{\mathrm{dz}}{}{\mathrm{dy}}}{{{y}}^{{2}}}$ (2.6)
 > $\mathrm{ContractIndices}\left(\mathrm{h1},\mathrm{CovariantDerivative}\left(\mathrm{CotTen1},\mathrm{Ch1}\right),\left[\left[1,1\right],\left[2,4\right]\right]\right)$
 ${0}{}{\mathrm{dx}}{}{\mathrm{dx}}$ (2.7)

Check that the Cotton tensor is a conformal invariant of the metric. We use the optional calling sequence in which the connection and curvature are specified.

 > $\mathrm{g2}≔f\left(x,z\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{g1}$
 ${\mathrm{g2}}{:=}{f}{}\left({x}{,}{z}\right){}{\mathrm{dx}}{}{\mathrm{dx}}{+}{f}{}\left({x}{,}{z}\right){}{x}{}{z}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{f}{}\left({x}{,}{z}\right){}{x}{}{z}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{f}{}\left({x}{,}{z}\right){}{{y}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}{f}{}\left({x}{,}{z}\right){}{{y}}^{{2}}{}{\mathrm{dz}}{}{\mathrm{dy}}$ (2.8)
 > $\mathrm{Ch2}≔\mathrm{Christoffel}\left(\mathrm{g2}\right):$
 > $\mathrm{R2}≔\mathrm{CurvatureTensor}\left(\mathrm{Ch2}\right):$
 > $\mathrm{CotTen2}≔\mathrm{CottonTensor}\left(\mathrm{g2},\mathrm{Ch2},\mathrm{R2}\right)$
 ${\mathrm{CotTen2}}{:=}{-}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{5}}}{+}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{5}}}{+}\frac{{x}{}\left({2}{}{z}{}{y}{+}{5}\right){}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{4}}}{-}\frac{{x}{}\left({2}{}{z}{}{y}{+}{5}\right){}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{4}}}{-}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dz}}}{{{y}}^{{3}}}{+}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dy}}}{{{y}}^{{3}}}$ (2.9)
 > $\mathrm{evalDG}\left(\mathrm{CotTen2}-\mathrm{CotTen1}\right)$
 ${0}$ (2.10)

Example 2.

We continue with the manifold and metric $\mathrm{g1}$ from Example 1. We check that the alternative form of the Cotton tensor is the dual of the default form of the tensor.

 M > $Y≔\mathrm{CottonTensor}\left(\mathrm{g1},\mathrm{indextype}=\left["con_bas","con_bas"\right]\right)$
 ${Y}{:=}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{D_x}}{}{\mathrm{D_z}}}{{{y}}^{{5}}}{+}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{D_z}}{}{\mathrm{D_x}}}{{{y}}^{{5}}}{-}\frac{{x}{}\left({{x}}^{{2}}{}{z}{+}{5}{}{y}\right){}{\mathrm{D_z}}{}{\mathrm{D_z}}}{{{y}}^{{7}}}$ (2.11)
 M > $\mathrm{CotUp}≔\mathrm{RaiseLowerIndices}\left(\mathrm{h1},\mathrm{CotTen1},\left[1\right]\right)$
 ${\mathrm{CotUp}}{:=}{-}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{5}}}{+}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{5}}}{+}\frac{{x}{}\left({{x}}^{{2}}{}{z}{+}{5}{}{y}\right){}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{{y}}^{{7}}}{-}\frac{{x}{}\left({{x}}^{{2}}{}{z}{+}{5}{}{y}\right){}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dx}}}{{{y}}^{{7}}}{-}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dz}}}{{{y}}^{{5}}}{+}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{D_z}}{}{\mathrm{dz}}{}{\mathrm{dy}}}{{{y}}^{{5}}}$ (2.12)
 M > $\mathrm{eps}≔\mathrm{PermutationSymbol}\left("con_bas"\right)$
 ${\mathrm{eps}}{:=}{\mathrm{D_x}}{}{\mathrm{D_y}}{}{\mathrm{D_z}}{-}{\mathrm{D_x}}{}{\mathrm{D_z}}{}{\mathrm{D_y}}{-}{\mathrm{D_y}}{}{\mathrm{D_x}}{}{\mathrm{D_z}}{+}{\mathrm{D_y}}{}{\mathrm{D_z}}{}{\mathrm{D_x}}{+}{\mathrm{D_z}}{}{\mathrm{D_x}}{}{\mathrm{D_y}}{-}{\mathrm{D_z}}{}{\mathrm{D_y}}{}{\mathrm{D_x}}$ (2.13)
 M > $Z≔\mathrm{ContractIndices}\left(\mathrm{eps},-\frac{1}{2}\mathrm{CotUp},\left[\left[2,2\right],\left[3,3\right]\right]\right)$
 ${Z}{:=}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{D_x}}{}{\mathrm{D_z}}}{{{y}}^{{5}}}{+}\frac{\left({-}{2}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{D_z}}{}{\mathrm{D_x}}}{{{y}}^{{5}}}{-}\frac{{x}{}\left({{x}}^{{2}}{}{z}{+}{5}{}{y}\right){}{\mathrm{D_z}}{}{\mathrm{D_z}}}{{{y}}^{{7}}}$ (2.14)
 M > $\mathrm{evalDG}\left(Y-Z\right)$
 ${0}$ (2.15) See Also