ProjectiveCurvatureTensor - Maple Help

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Tensor[ProjectiveCurvatureTensor] - calculate the Weyl projective curvature tensor of a connection on the tangent bundle

Calling Sequences

ProjectiveCurvature(g)

ProjectiveCurvature(C)

ProjectiveCurvature(R)

Parameters

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

C       - a connection on the tangent bundle of a manifold

R       - the curvature tensor of a connection on the tangent bundle of a manifold

Description

 • Let $C$ be a connection on the tangent bundle of a manifold $M$ of dimension$m>1.$ Let the curvature tensor of $C$ be $R$ and the Ricci tensor of $C$ be $Q$. The Weyl projective curvature of $C$ is the tensor $P$ of type $\left(\genfrac{}{}{0}{}{1}{3}\right)$ given by

 • With the first calling sequence the projective curvature tensor of the Christoffel connection of the metric g is computed. With the second calling sequence, the projective curvature tensor is computed directly from the given connection.  With the third calling sequence, the projective curvature tensor is computed directly from the given curvature tensor.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form ProjectiveCurvature(...) 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.

Compute the projective curvature of a metric.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],P\right)$
 ${\mathrm{frame name: P}}$ (2.1)
 P > $g≔\mathrm{evalDG}\left({ⅇ}^{\mathrm{λ}x}\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+\mathrm{dz}&t\mathrm{dz}\right)\right)$
 ${g}{:=}{{ⅇ}}^{{\mathrm{λ}}{}{x}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{{ⅇ}}^{{\mathrm{λ}}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{{ⅇ}}^{{\mathrm{λ}}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.2)
 P > $\mathrm{ProjectiveCurvatureTensor}\left(g\right)$
 $\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dx}}{-}\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}\frac{{1}}{{8}}{}{{\mathrm{λ}}}^{{2}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dy}}$ (2.3)

Example 2.

Compute the projective curvature of a connection.

 > $\mathrm{DGsetup}\left(\left[x,y,z,u\right],P\right)$
 ${\mathrm{frame name: P}}$ (2.4)
 P > $C≔\mathrm{Connection}\left(w\left(y\right)\left(\mathrm{D_x}&t\mathrm{dx}\right)&t\mathrm{dx}+f\left(u\right)\mathrm{D_y}&t\left(\mathrm{dx}&s\mathrm{dz}\right)\right)$
 ${C}{:=}{w}{}\left({y}\right){}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}\frac{{1}}{{2}}{}{f}{}\left({u}\right){}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{f}{}\left({u}\right){}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dx}}$ (2.5)
 P > $\mathrm{ProjectiveCurvatureTensor}\left(C\right)$
 ${-}\frac{{8}}{{15}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{8}}{{15}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}\frac{{1}}{{2}}{}{f}{}\left({u}\right){}{w}{}\left({y}\right){}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{f}{}\left({u}\right){}{w}{}\left({y}\right){}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dx}}{-}\frac{{1}}{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{u}}{}{f}{}\left({u}\right)\right){}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{du}}{+}\frac{{1}}{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{u}}{}{f}{}\left({u}\right)\right){}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{du}}{}{\mathrm{dz}}{+}\frac{{2}}{{15}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_y}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}\frac{{2}}{{15}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_y}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}\frac{{1}}{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{u}}{}{f}{}\left({u}\right)\right){}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{du}}{+}\frac{{1}}{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{u}}{}{f}{}\left({u}\right)\right){}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{du}}{}{\mathrm{dx}}{-}\frac{{4}}{{15}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{4}}{{15}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dy}}{-}\frac{{1}}{{15}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}\frac{{1}}{{15}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}\frac{{1}}{{5}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_z}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}\frac{{1}}{{5}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_z}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}\frac{{4}}{{15}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_u}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{du}}{+}\frac{{4}}{{15}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_u}}{}{\mathrm{dx}}{}{\mathrm{du}}{}{\mathrm{dy}}{-}\frac{{1}}{{15}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_u}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{du}}{+}\frac{{1}}{{15}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_u}}{}{\mathrm{dy}}{}{\mathrm{du}}{}{\mathrm{dx}}{+}\frac{{1}}{{5}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_u}}{}{\mathrm{du}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}\frac{{1}}{{5}}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}{}{w}{}\left({y}\right)\right){}{\mathrm{D_u}}{}{\mathrm{du}}{}{\mathrm{dy}}{}{\mathrm{dx}}$ (2.6)

Example 3.

Compute the projective curvature from a given curvature tensor.

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.7)
 M > $g≔\mathrm{evalDG}\left(-\mathrm{dt}&t\mathrm{dt}+{ⅇ}^{a\left(t\right)}\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+\mathrm{dz}&t\mathrm{dz}\right)\right)$
 ${g}{:=}{-}{\mathrm{dt}}{}{\mathrm{dt}}{+}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.8)
 M > $R≔\mathrm{CurvatureTensor}\left(g\right):$
 M > $\mathrm{ProjectiveCurvatureTensor}\left(R\right)$
 $\frac{{1}}{{3}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{-}\frac{{1}}{{3}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}{+}\frac{{1}}{{3}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_t}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dy}}{-}\frac{{1}}{{3}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_t}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dt}}{+}\frac{{1}}{{3}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_t}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}\frac{{1}}{{3}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_t}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dt}}{-}\frac{{1}}{{6}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{1}}{{6}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}\frac{{1}}{{6}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}\frac{{1}}{{6}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}\frac{{1}}{{6}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}\frac{{1}}{{6}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}\frac{{1}}{{6}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{1}}{{6}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}\frac{{1}}{{6}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}\frac{{1}}{{6}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}\frac{{1}}{{6}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}\frac{{1}}{{6}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{a}{}\left({t}\right)\right)\right){}{{ⅇ}}^{{a}{}\left({t}\right)}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dy}}$ (2.9)