 ParallelTransportEquations - Maple Help

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Tensor[ParallelTransportEquations] - calculate the parallel transport equations for a linear connection on the tangent bundle or a linear connection on a vector bundle

Calling Sequences

ParallelTransportEquations(C, Y, ${\mathbf{Γ}},$ t)

Parameters

C       - a list of functions of a single variable, defining the components of a curve on a manifold $M$, with respect to a given coordinate system

Y       - a vector field defined along the curve $C$

$\mathrm{Γ}$       - a connection on the tangent bundle to a manifold $M$ or a connection on a vector bundle $E\to M$

t       - the curve parameter Description

 • Let $M$ be a manifold and let $\nabla$ be a linear connection on the tangent bundle of $M$ or a connection on a vector bundle $E\to M$. If $C$ is a curve in $M$ with tangent vector $T$, then the parallel transport equations for a vector field $Y$ along $C$ are the linear, first order ODEs defined by ${\nabla }_{T}Y=0$.
 • The procedure ParallelTransportEquations(C, Y, ${\mathbf{\Gamma }}$, t) returns the vector ${\nabla }_{T}Y$.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form ParallelTransportEquations(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-ParallelTransportEquations. 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 > $\mathrm{Gamma}≔\mathrm{Connection}\left(-\left(\mathrm{D_x}&t\mathrm{dx}\right)&t\mathrm{dy}+\left(\mathrm{D_y}&t\mathrm{dy}\right)&t\mathrm{dx}\right)$
 ${\mathrm{Γ}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{2}{,}{1}\right]{,}{1}\right]\right]\right]\right)$ (2.2)

To define the parallel transport equations along $C$, we first define a curve on $M$ by specifying a list of functions of a single variable $t$. We also define a vector field $Y$ with coefficients depending on the curve parameter.

 M > $C≔\left[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)\right]$
 ${C}{:=}\left[{\mathrm{cos}}{}\left({t}\right){,}{\mathrm{sin}}{}\left({t}\right)\right]$ (2.3)
 M > $Y≔\mathrm{evalDG}\left(A\left(t\right)\mathrm{D_x}+B\left(t\right)\mathrm{D_y}\right)$
 ${Y}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{A}{}\left({t}\right)\right]{,}\left[\left[{2}\right]{,}{B}{}\left({t}\right)\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{A}{}\left({t}\right)\right]{,}\left[\left[{2}\right]{,}{B}{}\left({t}\right)\right]\right]\right]\right)$ (2.4)

The program ParallelTransportEquations returns a vector whose components define the parallel transport equations.

 M > $V≔\mathrm{ParallelTransportEquations}\left(C,Y,\mathrm{Gamma},t\right)$
 ${V}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{A}{}\left({t}\right){}{\mathrm{cos}}{}\left({t}\right){+}\stackrel{{\mathbf{.}}}{{A}}{}\left({t}\right)\right]{,}\left[\left[{2}\right]{,}{-}{B}{}\left({t}\right){}{\mathrm{sin}}{}\left({t}\right){+}\stackrel{{\mathbf{.}}}{{B}}{}\left({t}\right)\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{A}{}\left({t}\right){}{\mathrm{cos}}{}\left({t}\right){+}\stackrel{{\mathbf{.}}}{{A}}{}\left({t}\right)\right]{,}\left[\left[{2}\right]{,}{-}{B}{}\left({t}\right){}{\mathrm{sin}}{}\left({t}\right){+}\stackrel{{\mathbf{.}}}{{B}}{}\left({t}\right)\right]\right]\right]\right)$ (2.5)

To solve these parallel transport equations use the DGinfo command in the Tools package to obtain the coefficients of $V$ as a set. Pass the resulting system of 1st order ODEs to dsolve.

 M > $\mathrm{DE}≔\mathrm{Tools}:-\mathrm{DGinfo}\left(V,"CoefficientSet"\right)$
 ${\mathrm{DE}}{:=}\left\{{-}{A}{}\left({t}\right){}{\mathrm{cos}}{}\left({t}\right){+}\frac{{ⅆ}}{{ⅆ}{t}}{}{A}{}\left({t}\right){,}{-}{B}{}\left({t}\right){}{\mathrm{sin}}{}\left({t}\right){+}\frac{{ⅆ}}{{ⅆ}{t}}{}{B}{}\left({t}\right)\right\}$ (2.6)
 M > $\mathrm{soln}≔\mathrm{dsolve}\left(\mathrm{DE},\mathrm{explicit}\right)$
 ${\mathrm{soln}}{:=}\left\{{A}{}\left({t}\right){=}{\mathrm{_C2}}{}{{ⅇ}}^{{\mathrm{sin}}{}\left({t}\right)}{,}{B}{}\left({t}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{\mathrm{cos}}{}\left({t}\right)}\right\}$ (2.7)

Back substitute the solution into the vector field $Y$.

 M > $\mathrm{Y_t}≔\mathrm{eval}\left(Y,\mathrm{soln}\right)$
 ${\mathrm{Y_t}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{\mathrm{_C2}}{}{{ⅇ}}^{{\mathrm{sin}}{}\left({t}\right)}\right]{,}\left[\left[{2}\right]{,}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{\mathrm{cos}}{}\left({t}\right)}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{\mathrm{_C2}}{}{{ⅇ}}^{{\mathrm{sin}}{}\left({t}\right)}\right]{,}\left[\left[{2}\right]{,}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{\mathrm{cos}}{}\left({t}\right)}\right]\right]\right]\right)$ (2.8)

Example 2.

First create a rank 2 vector bundle $E\to M$ over the two-dimensional manifold $M$ and define a connection on $E$.

 M > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u,v\right],E\right):$
 E > $\mathrm{Gamma}≔\mathrm{Connection}\left(\left(\mathrm{D_v}&t\mathrm{dv}\right)&t\mathrm{dy}-\left(\mathrm{D_u}&t\mathrm{dv}\right)&t\mathrm{dx}\right)$
 ${\mathrm{Γ}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{E}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{3}{,}{4}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{E}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{3}{,}{4}{,}{1}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}{,}{2}\right]{,}{1}\right]\right]\right]\right)$ (2.9)

Define a curve $C$ in $M$.

 E > $C≔\left[t,t\right]$
 ${C}{:=}\left[{t}{,}{t}\right]$ (2.10)
 E > $Y≔\mathrm{evalDG}\left(A\left(t\right)\mathrm{D_u}+B\left(t\right)\mathrm{D_v}\right)$
 ${Y}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{A}{}\left({t}\right)\right]{,}\left[\left[{4}\right]{,}{B}{}\left({t}\right)\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{A}{}\left({t}\right)\right]{,}\left[\left[{4}\right]{,}{B}{}\left({t}\right)\right]\right]\right]\right)$ (2.11)

The program ParallelTransportEquations returns a vector whose components define the parallel transport equations.

 E > $V≔\mathrm{ParallelTransportEquations}\left(C,Y,\mathrm{Gamma},t\right)$
 ${V}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{-}{B}{}\left({t}\right){+}\stackrel{{\mathbf{.}}}{{A}}{}\left({t}\right)\right]{,}\left[\left[{4}\right]{,}{B}{}\left({t}\right){+}\stackrel{{\mathbf{.}}}{{B}}{}\left({t}\right)\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{-}{B}{}\left({t}\right){+}\stackrel{{\mathbf{.}}}{{A}}{}\left({t}\right)\right]{,}\left[\left[{4}\right]{,}{B}{}\left({t}\right){+}\stackrel{{\mathbf{.}}}{{B}}{}\left({t}\right)\right]\right]\right]\right)$ (2.12)

To solve these parallel transport equations use DGinfo  to obtain the coefficients of as a set. Pass the result to dsolve.

 E > $\mathrm{DE}≔\mathrm{Tools}:-\mathrm{DGinfo}\left(V,"CoefficientSet"\right)$
 ${\mathrm{DE}}{:=}\left\{{-}{B}{}\left({t}\right){+}\frac{{ⅆ}}{{ⅆ}{t}}{}{A}{}\left({t}\right){,}{B}{}\left({t}\right){+}\frac{{ⅆ}}{{ⅆ}{t}}{}{B}{}\left({t}\right)\right\}$ (2.13)
 E > $\mathrm{soln}≔\mathrm{dsolve}\left(\mathrm{DE},\mathrm{explicit}\right)$
 ${\mathrm{soln}}{:=}\left\{{A}{}\left({t}\right){=}{-}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{t}}{+}{\mathrm{_C1}}{,}{B}{}\left({t}\right){=}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{t}}\right\}$ (2.14)

Back substitute the solution into the vector field $Y$.

 E > $\mathrm{Y_t}≔\mathrm{eval}\left(Y,\mathrm{soln}\right)$
 ${\mathrm{Y_t}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{-}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{t}}{+}{\mathrm{_C1}}\right]{,}\left[\left[{4}\right]{,}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{t}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{-}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{t}}{+}{\mathrm{_C1}}\right]{,}\left[\left[{4}\right]{,}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{t}}\right]\right]\right]\right)$ (2.15) See Also