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 calculates curvature tensor. The procedure is only available if the curvature 2-forms have been calculated (see atlas[Curvature] ).
• The Ricci  procedure calculates Ricci tensor. The procedure is only available if the curvature 2-forms (see atlas[Curvature] ) has been calculated.
• The RicciScalar  procedure calculates Ricci scalar. The procedure is only available if 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);

Declare vectors:
Vectors(X,Y,Z,E[j]);

Declare constant :
Constants(lambda);

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)));

Declare frame:
Frame(E[i]);

d(x);

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

Connection calculation:
Connection(omega);

Curvature calculation:
Curvature(Omega);

Riemannian tensor calculation:
Riemann(R);

Ricci tensor calculation:
Ricci(r);

Ricci scalar calculation:
RicciScalar(s);

Example 2
restart:
with(atlas):

Declare forms:
Forms(e[j]=1,xi=1);

Declare vectors:
Vectors(X,Y,Z,E[j]);

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

Declare frame:
Frame(E[i]);

Connection definition:
omega[1,1]:=x*e[1];

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

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

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

Connection declaration:
Connection(omega);

Curvature calculation:
Curvature(Omega);

Riemann(R);

Ricci calculation:
Ricci(r);

RicciScalar(s);

See Also:

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