Benoit in Waterloo

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These are questions asked by Benoit in Waterloo

The help for the package Differential Geometry suggests one way to create an anholonomic frame, but the method suggested assumes one knows the frame and can write it in the coordinates of the manifold, and use this representation to compute the structure equations. It is useful to turn this problem around: suppose I know the structure equations but I do not know the frame, how do I correctly enter this in Maple so that later computations are correct?

 

Here is an example:

with(DifferentialGeometry): with(LieAlgebras):
DGsetup([x, y], R2):          
Fr := [A, B]; StructureEquations := [[A, B] = a(x, y)*A];
LD := LieAlgebras:-LieAlgebraData(StructureEquations, Fr, M);
DGsetup(LD, verbose):

that creates the Lie algebra M. I thought it would produce the manifold M whose frame {e1,e2} (or {A,B})) obeys the structure equations. But it does not. Indeed, the simple example

LieBracket(e1, f(x, y)*e2);

returns

f(x,y) a(x,y) e1.

The extra term where e1 differentiates f(x,y) does not appear. Just to be clear, the correct answer is

f(x,y) a(x,y) e1 + e1[f] e2.

 

How should I correctly setup this manifold?

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