 SchoutenTensor - Maple Help

Tensor[SchoutenTensor] - calculate the Schouten tensor of a metric

Calling Sequences

SchoutenTensor(g)

SchoutenTensor(g, R)

Parameters

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

R       - (optional) the curvature tensor of g Description

 • Let ${g}_{\mathrm{ab}}$ be metric (of any signature) on the tangent bundle of a manifold $M$ of dimension$n>2.$ Let the Ricci tensor of $g$ be ${R}_{\mathrm{ab}}$ with scalar curvature $R={g}^{\mathrm{ab}}{R}_{\mathrm{ab}}$. The the Schouten tensor of ${g}_{\mathrm{ab}}$ is the symmetric tensor

${S}_{\mathrm{ab}}=\frac{1}{n-2}\left({R}_{\mathrm{ab}}-\frac{R}{2\left(n-1\right)}{g}_{\mathrm{ab}}\right).$

 • The first calling sequence computes the curvature tensor, Ricci tensor, and Ricci Scalar directly from the given metric. The second calling sequence uses the given curvature tensor  to compute the Ricci tensor via and then computes the scalar curvature using the given metric via
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SchoutenTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-ProjectiveCurvature. Examples

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

Example 1.

Calculate the Schouten tensor of a metric.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],P\right)$
 ${\mathrm{frame name: P}}$ (2.1)
 P > $g≔\mathrm{evalDG}\left(\mathrm{exp}\left(\mathrm{\lambda }x\right)\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)\right)$
 ${g}{:=}{{ⅇ}}^{{\mathrm{λ}}{}{x}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{{ⅇ}}^{{\mathrm{λ}}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{{ⅇ}}^{{\mathrm{λ}}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.2)
 P > $\mathrm{SchoutenTensor}\left(g\right)$
 $\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}{-}\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.3)

Example 2.

Calculate the Schouten tensor from a metric and curvature tensor.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],P\right)$
 ${\mathrm{frame name: P}}$ (2.4)
 P > $g≔\mathrm{evalDG}\left(\mathrm{exp}\left(\mathrm{\lambda }x\right)\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)\right)$
 ${g}{:=}{{ⅇ}}^{{\mathrm{λ}}{}{x}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{{ⅇ}}^{{\mathrm{λ}}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{{ⅇ}}^{{\mathrm{λ}}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.5)
 P > $R≔\mathrm{CurvatureTensor}\left(g\right)$
 ${R}{:=}{-}\frac{{1}}{{4}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{1}}{{4}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}\frac{{1}}{{4}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}\frac{{1}}{{4}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dy}}$ (2.6)
 P > $\mathrm{SchoutenTensor}\left(g,R\right)$
 $\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}{-}\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.7)