atlas[Curvature] - calculation of curvature 2-forms 

Calling Sequence: 

    Curvature(Id) 

Parameters: 

    Id - variable - curvature identifier 

Description: 

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

 

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

 

Examples: 

Example 1 

Conformally flat metric of sphere -  

>	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 constant :
Constants(lambda); 

(2.1.3)

 

Declare the 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))); 

(2.1.4)

 

Declare the frame:
Frame(E[i]); 

(2.1.5)

 

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

(2.1.6)

 

Connection calculation: 

>	Connection(omega);

 

(2.1.7)

 

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

 

(2.1.8)

 

Curvature calculation: 

>	Curvature(Omega);

 

(2.1.9)

 

Now curvature 2-forms are calculated: 

>	eval(Omega);

 

(2.1.10)

 

>	'Omega[1,3]'=Omega[1,3];

 

(2.1.11)

 

>	'iota[E[k]](Omega[2,1])'=iota[E[k]](Omega[2,1]);

 

(2.1.12)

 

Example 2 

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

(2.2.3)

 

Declare frame:
Frame(E[i]); 

(2.2.4)

 

Connection definition: 

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

 

(2.2.5)

 

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

 

(2.2.6)

 

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

 

(2.2.7)

 

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

 

(2.2.8)

 

Connection declaration: 

>	Connection(omega);

 

(2.2.9)

 

Curvature calculation:
Curvature(Omega); 

(2.2.10)

 

>	eval(Omega);

 

(2.2.11)

 

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

 

(2.2.12)

 

>

 

See Also:  

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