 calculate the torsion tensor for a linear connection on the tangent bundle - Maple Programming Help

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Tensor[TorsionTensor] - calculate the torsion tensor for a linear connection on the tangent bundle

Calling Sequences

TorsionTensor(C)

Parameters

C    - a connection on the tangent bundle to a manifold

Description

 • Let $M$ be a manifold and let $\nabla$ be a linear connection on the tangent bundle of $M$. The torsion tensor $S$ of $\nabla$ is the rank 3 tensor (tensor of type$\left(\genfrac{}{}{0}{}{1}{2}\right)$) defined by  .  Here $X,Y$ are vector fields on $M$.
 • The connection $\nabla$ is said to be symmetric if its torsion tensor $S$ vanishes.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form TorsionTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-TorsionTensor.

Examples

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

Example 1.

First create a 2 dimensional manifold $M$ and define a connection on the tangent space of $M$.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)
 M > $\mathrm{C1}≔\mathrm{Connection}\left(a\left(\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+b\left(\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)$
 ${\mathrm{C1}}{:=}{a}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{b}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.2)
 M > $\mathrm{TorsionTensor}\left(\mathrm{C1}\right)$
 ${-}{a}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{a}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dx}}$ (2.3)

Example 2.

Define a frame on $M$ and use this frame to specify a connection $\mathrm{C2}$ on the tangent space of $M$.  While the connection $\mathrm{C2}$ is "symmetric" in its covariant indices, it is not a symmetric connection.

 M > $\mathrm{FR}≔\mathrm{FrameData}\left(\left[\frac{1}{y}\mathrm{dx},\frac{1}{x}\mathrm{dy}\right],\mathrm{M1}\right):$
 M > $\mathrm{DGsetup}\left(\mathrm{FR}\right)$
 ${\mathrm{frame name: M1}}$ (2.4)
 M1 > $\mathrm{C2}≔\mathrm{Connection}\left(\left(\mathrm{E1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{Θ2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{Θ2}\right)$
 ${\mathrm{C2}}{:=}{\mathrm{E1}}{}{\mathrm{Θ2}}{}{\mathrm{Θ2}}$ (2.5)
 M1 > $\mathrm{TorsionTensor}\left(\mathrm{C2}\right)$
 $\frac{{x}{}{\mathrm{E1}}{}{\mathrm{Θ1}}{}{\mathrm{Θ2}}}{{y}}{-}\frac{{x}{}{\mathrm{E1}}{}{\mathrm{Θ2}}{}{\mathrm{Θ1}}}{{y}}{-}\frac{{y}{}{\mathrm{E2}}{}{\mathrm{Θ1}}{}{\mathrm{Θ2}}}{{x}}{+}\frac{{y}{}{\mathrm{E2}}{}{\mathrm{Θ2}}{}{\mathrm{Θ1}}}{{x}}$ (2.6)