Connection with torsion

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Problem:

For given connection  calculate the following :

• curvature 2-forms
• torsion 2-forms
• Riemannian tensor
• Ricci tensor

restart:
with(atlas):

Function declaration:
Functions(f=f(x,y),h=h(x,y),g=g(x,y),z=z(x,y));

Vector fields:
Vectors(E[i],X,Y,Z);

Differential p-forms:
Forms(e[j]=1);

Coframe 1-forms:
Coframe(e[1]=d(x),e[2]=d(y));

Frame vector fields:
Frame(E[i]);

Connection definition:
omega[1,1]:=f*e[2];

omega[1,2]:=0;

omega[2,1]:=0;

omega[2,2]:=h*e[1];

Connection declaration:
Connection(omega);

Curvature calculation:
Curvature(Omega);

Result:
eval(Omega);

Torsion calculation:
Torsion(T);

Result:
eval(T);

Curvature tensor calculation:
Riemann(R);

Ricci tensor calculation:
Ricci(r);

Some more simple calculations:

Covariant derivatives:
'cov(E[j],e[1])'=cov(E[j],e[1]);
'cov(E[j],e[2])'=cov(E[j],e[2]);

Lie derivative:
'L[E[1]](E[2])'=L[E[1]](E[2]);

Interior products:
'iota[E[k]](T[1])'=iota[E[k]](T[1]);
'iota[E[k]](Omega[1,1])'=iota[E[k]](Omega[1,1]);