Connection with torsion 

Description: 

This worksheet illustrates how to use the atlas package to make calculations with user defined connections.  

 

>	restart: with(atlas):

 

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

(2.1)

 

Vector fields:
Vectors(E[i],X,Y,Z); 

(2.2)

 

Differential p-forms:
Forms(e[j]=1); 

(2.3)

 

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

(2.4)

 

Frame vector fields:
Frame(E[i]); 

(2.5)

 

Connection definition: 

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

 

(2.6)

 

>	omega[1,2]:=0;

 

(2.7)

 

>	omega[2,1]:=0;

 

(2.8)

 

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

 

(2.9)

 

Connection declaration:
Connection(omega); 

(2.10)

 

Curvature calculation:
Curvature(Omega); 

(2.11)

 

Result:
eval(Omega); 

(2.12)

 

Torsion calculation:
Torsion(T); 

(2.13)

 

Result:
eval(T); 

(2.14)

 

Curvature tensor calculation:
Riemann(R); 

(2.15)

 

Ricci tensor calculation:
Ricci(r); 

(2.16)

 

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]); 

 

	(2.17)

 

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

(2.18)

 

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

 

	(2.19)

 

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