RicciScalar - Maple Help
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Tensor[RicciScalar] - calculate the Ricci scalar for a metric

Calling Sequences

RicciScalar(g, R)

Parameters

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

R    - (optional) the curvature tensor of the metric $g$ calculated from the Christoffel symbol of $g$

Description

 • The Ricci scalar $S$ for a metric $g$ is the total contraction of the inverse of $g$ with the Ricci tensor $R$ of $g$. In components, $S={g}^{\mathrm{ab}}{R}_{\mathrm{ab}}.$
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form RicciScalar(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-RicciScalar.

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 $\mathrm{g1}$ on $M$.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)
 M > $\mathrm{g1}≔\mathrm{evalDG}\left(\frac{{a}^{2}\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+\mathrm{dz}&t\mathrm{dz}\right)}{{\left({k}^{2}+{x}^{2}+{y}^{2}+{z}^{2}\right)}^{2}}\right)$
 ${\mathrm{g1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{{a}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{{a}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{{a}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{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]{,}\frac{{{a}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{{a}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{{a}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{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]{,}\frac{{{a}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{{a}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{{a}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{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]{,}\frac{{{a}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{{a}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{{a}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]\right]\right]\right)$ (2.2)
 M > $\mathrm{C1}≔\mathrm{Christoffel}\left(\mathrm{g1}\right):$

Calculate the curvature tensor.

 M > $\mathrm{R1}≔\mathrm{CurvatureTensor}\left(\mathrm{C1}\right)$
 ${\mathrm{R1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{1}{,}{2}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}{,}{1}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{1}{,}{3}{,}{1}{,}{3}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{1}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{3}{,}{2}{,}{3}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{2}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{2}{,}{2}{,}{3}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{2}{,}{3}{,}{2}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{1}{,}{2}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}{,}{1}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{1}{,}{3}{,}{1}{,}{3}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{1}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{3}{,}{2}{,}{3}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{2}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{2}{,}{2}{,}{3}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{2}{,}{3}{,}{2}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{1}{,}{2}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}{,}{1}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{1}{,}{3}{,}{1}{,}{3}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{1}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{3}{,}{2}{,}{3}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{2}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{2}{,}{2}{,}{3}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{2}{,}{3}{,}{2}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{1}{,}{2}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{1}{,}{2}{,}{2}{,}{1}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{1}{,}{3}{,}{1}{,}{3}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{1}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{3}{,}{2}{,}{3}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{2}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{2}{,}{2}{,}{3}\right]{,}{-}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}{,}{2}{,}{3}{,}{2}\right]{,}\frac{{4}{}{{k}}^{{2}}}{{\left({{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{2}}}\right]\right]\right]\right)$ (2.3)

Calculate the Ricci scalar.

 M > $\mathrm{S1}≔\mathrm{RicciScalar}\left(\mathrm{g1},\mathrm{R1}\right)$
 ${\mathrm{S1}}{:=}\frac{{24}{}{{k}}^{{2}}}{{{a}}^{{2}}}$ (2.4)

Example 2.

We re-work the previous example in an orthonormal frame.

 M > $f≔\frac{a}{{k}^{2}+{x}^{2}+{y}^{2}+{z}^{2}}$
 ${f}{:=}\frac{{a}}{{{k}}^{{2}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}}$ (2.5)
 M > $\mathrm{FR}≔\mathrm{FrameData}\left(\left[f\mathrm{dx},f\mathrm{dy},f\mathrm{dz}\right],\mathrm{M1}\right):$
 M > $\mathrm{DGsetup}\left(\mathrm{FR}\right)$
 ${\mathrm{frame name: M1}}$ (2.6)
 M1 > $\mathrm{g3}≔\mathrm{evalDG}\left(\mathrm{Θ1}&t\mathrm{Θ1}+\mathrm{Θ2}&t\mathrm{Θ2}+\mathrm{Θ3}&t\mathrm{Θ3}\right)$
 ${\mathrm{g3}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{M1}}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{M1}}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{M1}}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{M1}}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)$ (2.7)

Calculate the Ricci scalar.

 M1 > $\mathrm{S3}≔\mathrm{RicciScalar}\left(\mathrm{g3}\right)$
 ${\mathrm{S3}}{:=}\frac{{24}{}{{k}}^{{2}}}{{{a}}^{{2}}}$ (2.8)