atlas[Curvature] - calculation of curvature 2-forms

Calling Sequence:

     Curvature(Id)

Parameters:

      Id - variable - curvature identifier

Description:

• The Curvature  procedure calculates curvature 2-forms. If a connection is calculated or defined then curvature can be calculated completely otherwise just curvature identifier is declared as a curvature. To get the rezult of the calculation use eval  or atlas[iota]  operators.
• The definition is as follows:  , where  are connection 1-forms,  is the dimension.

Example 1

Conformally flat metric of 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]);

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

Connection calculation:
Connection(omega);

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

Curvature calculation:
Curvature(Omega);

Now curvature 2-forms  are calculated:
eval(Omega);

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

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

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);

eval(Omega);

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

See Also:

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