atlas[Connection] - calculation of connection 1-forms 

Calling Sequence: 

    Connection(Id) 

Parameters: 

    Id - variable - connection identifier 

Description: 

The Connection procedure allows one to calculate or define connection 1-forms. If a metric is defined, then the connection is calculated with respect to the metric, otherwise just the connection identifier is declared as a connection. If the identifier is a table then the table is declared as a connection (see example 3). To get the result of the calculation use eval or atlas[iota] operators. 

Examples: 

Example 1 

Abstract connection  

>	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 with 1-forms: (see atlas[Coframe]) 

>	Coframe(e[k],k=1..n);

 

(2.1.3)

 

Declare frame vectors (see atlas[Frame]): 

>	Frame(E[i]);

 

(2.1.4)

 

Declaration of connection 1-forms:  

>	Connection(omega);

 

(2.1.5)

 

>	Torsion(T);

 

(2.1.6)

 

>	'd(e[j])'=d(e[j]);

 

(2.1.7)

 

>	d(xi)=d(ToBasis(xi));

 

(2.1.8)

 

Example 2 

Conformally flat metric of sphere -  

>	restart: with(atlas):

 

Declare forms: 

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

 

(2.2.1)

 

Declare vectors: 

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

 

(2.2.2)

 

Declare constant :
Constants(lambda); 

(2.2.3)

 

Declare coframe:
Coframe(e[1]=2*d(x)/(1+lambda*(x^2+y^2+z^2)),e[2]=2*d(y)/(1+lambda*(x^2+y^2+z^2)),e[3]=2*d(z)/(1+lambda*(x^2+y^2+z^2))); 

(2.2.4)

 

Declare frame:
Frame(E[i]); 

(2.2.5)

 

Basis one forms are not exact:
'd(e[1])'=normal(d(e[1])); 

(2.2.6)

 

Declare conformally flat metric of   (see atlas[Metric]):
Metric(g=e[1]&.e[1]+e[2]&.e[2]+e[3]&.e[3]); 

(2.2.7)

 

Connection calculation: 

>	Connection(omega);

 

(2.2.8)

 

Now connection 1-forms are calculated: 

>	eval(omega);

 

(2.2.9)

 

>	'iota[E[i]](omega[1,2])'=iota[E[i]](omega[1,2]);

 

(2.2.10)

 

Example 3 

>	restart: with(atlas):

 

Declare forms: 

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

 

(2.3.1)

 

Declare vectors: 

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

 

(2.3.2)

 

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

(2.3.3)

 

Declare frame:
Frame(E[i]); 

(2.3.4)

 

Connection definition: 

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

 

(2.3.5)

 

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

 

(2.3.6)

 

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

 

(2.3.7)

 

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

 

(2.3.8)

 

Connection declaration: 

>	Connection(omega);

 

(2.3.9)

 

Curvature calculation:
Curvature(Omega); 

(2.3.10)

 

>	eval(Omega);

 

(2.3.11)

 

Torsion calculation:
Torsion(T); 

(2.3.12)

 

>

eval(T);

 

(2.3.13)

 

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

 

(2.3.14)

 

>

 

See Also:  

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