Example 1.

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

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.

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

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.

Back substitute the solution into the vector field Y.

Example 2.

First create a rank 2 vector bundle E -> M dimensional manifold M and define a connection on E.

Define a curve C in M.

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

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

Back substitute the solution into the vector field Y.

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