atlas[Riemann] - calculation of Riemannian tensor
 

atlas[Ricci] - calculation of Ricci tensor
 

atlas[RicciScalar] - calculation of Ricci scalar 

Calling Sequence: 

    Riemann(Id)
    Ricci(Id)
    RicciScalar(Id) 

Parameters: 

    Id - variable - corresponding identifier 

Description: 

• The Riemann procedure allows one to calculate the curvature tensor. The procedure is only available if the curvature 2-forms have been calculated (see atlas[Curvature]).

 

• The Ricci procedure allows one to calculate the Ricci tensor. The procedure is only available if the curvature 2-forms (see atlas[Curvature]) have been calculated.

 

• The RicciScalar procedure allows one to calculate the Ricci scalar. The procedure is only available if the metric tensor is definite (see atlas[Metric]) and the Ricci tensor has been calculated.

 

Examples: 

3-dimensional 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 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 frame:
Frame(E[i]); 

(2.1.5)

 

>

d(x);

 

(2.1.6)

 

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

(2.1.7)

 

Connection calculation: 

>	Connection(omega);

 

(2.1.8)

 

Curvature calculation: 

>	Curvature(Omega);

 

(2.1.9)

 

Riemannian tensor calculation: 

>	Riemann(R);

 

(2.1.10)

 

Ricci tensor calculation: 

>

Ricci(r);

 

(2.1.11)

 

Ricci scalar calculation: 

>	RicciScalar(s);

 

(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)

 

>	Riemann(R);

 

(2.2.11)

 

Ricci calculation:
Ricci(r); 

(2.2.12)

 

>	RicciScalar(s);

 

Warning, There is no actual metric tensor

 

>

 

See Also:  

atlas, atlas[Connection], atlas[Curvature], atlas[Metric].