atlas[Torsion] - calculation of torsion 2-forms 

Calling Sequence: 

    Torsion(Id) 

Parameters: 

    Id - variable - torsion identifier 

Description: 

• The Torsion procedure allows one to calculate torsion 2-forms. If a connection is calculated or defined, then the torsion can be calculated completely. Otherwise just the torsion identifier is declared as the torsion. To get the result of the calculation use eval or atlas[iota] operators.

 

• The definition is as follows: , where are coframe 1-forms,   are connection 1-forms, is the dimension.

 

Examples: 

Example 1 

>	restart: with(atlas):

 

Declare forms: 

>	Forms(e[j]=1,xi=1);

 

(2.1.1)

 

Declare vectors: 

>	Vectors(X,Y,Z,E[j]);

 

(2.1.2)

 

Declare coframe:
Coframe(e[1]=x*d(x)+y*d(y),e[2]=x*d(y)-y*d(x)); 

(2.1.3)

 

Declare frame:
Frame(E[i]); 

(2.1.4)

 

Connection definition: 

>	omega[1,1]:=x*e[1];

 

(2.1.5)

 

>	omega[2,2]:=y*e[2];

 

(2.1.6)

 

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

 

(2.1.7)

 

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

 

(2.1.8)

 

Connection declaration: 

>	Connection(omega);

 

(2.1.9)

 

Torsion calculation:
Torsion(Omega); 

(2.1.10)

 

>	eval(Omega);

 

(2.1.11)

 

>	'L[E[1]](E[2])'=L[E[1]](E[2]);

 

(2.1.12)

 

>

 

See Also:  

atlas, atlas[Frame], atlas[Coframe], atlas[Metric].