ParallelTransportEquations - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : DifferentialGeometry : Tensor : DifferentialGeometry/Tensor/ParallelTransportEquations

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, 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

   Γ       - a connection on the tangent bundle to a manifold M or a connection on a vector bundle EM

   t       - the curve parameter

 

Description

Examples

See Also

Description

• 

Let M be a manifold and let  be a linear connection on the tangent bundle of M or a connection on a vector bundle EM. 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 TY=0.

• 

The procedure ParallelTransportEquations(C, Y, Γ, t) returns the vector TY.

• 

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

withDifferentialGeometry:withTensor:

 

Example 1.

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

DGsetupx,y,M

frame name: M

(2.1)
M > 

GammaConnectionD_x&tdx&tdy+D_y&tdy&tdx

Γ:=_DGconnection,M,con_bas,cov_bas,cov_bas,,1,1,2,−1,2,2,1,1,_DGconnection,M,con_bas,cov_bas,cov_bas,,1,1,2,−1,2,2,1,1

(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 > 

Ccost,sint

C:=cost,sint

(2.3)
M > 

YevalDGAtD_x+BtD_y

Y:=_DGvector,M,,1,At,2,Bt,_DGvector,M,,1,At,2,Bt

(2.4)

 

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

M > 

VParallelTransportEquationsC,Y,Gamma,t

V:=_DGvector,M,,1,Atcost+A.t,2,Btsint+B.t,_DGvector,M,,1,Atcost+A.t,2,Btsint+B.t

(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 > 

DETools:-DGinfoV,CoefficientSet

DE:=Atcost+ⅆⅆtAt,Btsint+ⅆⅆtBt

(2.6)
M > 

solndsolveDE,explicit

soln:=At=_C2ⅇsint,Bt=_C1ⅇcost

(2.7)

 

Back substitute the solution into the vector field Y.

M > 

Y_tevalY,soln

Y_t:=_DGvector,M,,1,_C2ⅇsint,2,_C1ⅇcost,_DGvector,M,,1,_C2ⅇsint,2,_C1ⅇcost

(2.8)

 

Example 2.

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

M > 

DGsetupx,y,u,v,E:

E > 

GammaConnectionD_v&tdv&tdyD_u&tdv&tdx

Γ:=_DGconnection,E,con_vrt,cov_vrt,cov_bas,,3,4,1,−1,4,4,2,1,_DGconnection,E,con_vrt,cov_vrt,cov_bas,,3,4,1,−1,4,4,2,1

(2.9)

Define a curve C in M.

E > 

Ct,t

C:=t,t

(2.10)
E > 

YevalDGAtD_u+BtD_v

Y:=_DGvector,E,,3,At,4,Bt,_DGvector,E,,3,At,4,Bt

(2.11)

 

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

E > 

VParallelTransportEquationsC,Y,Gamma,t

V:=_DGvector,E,,3,Bt+A.t,4,Bt+B.t,_DGvector,E,,3,Bt+A.t,4,Bt+B.t

(2.12)

 

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

E > 

DETools:-DGinfoV,CoefficientSet

DE:=Bt+ⅆⅆtAt,Bt+ⅆⅆtBt

(2.13)
E > 

solndsolveDE,explicit

soln:=At=_C2ⅇt+_C1,Bt=_C2ⅇt

(2.14)

 

Back substitute the solution into the vector field Y.

E > 

Y_tevalY,soln

Y_t:=_DGvector,E,,3,_C2ⅇt+_C1,4,_C2ⅇt,_DGvector,E,,3,_C2ⅇt+_C1,4,_C2ⅇt

(2.15)

See Also

DifferentialGeometry, Tensor, Christoffel, Connection, CovariantDerivative, DGinfo, DirectionalCovariantDerivative