Connection with torsion

Copyright © 2003-2010 by  DigiArea Group . All rights reserved.

Description:

This worksheet illustrates how to use atlas  package to make calculations with user defined connection. First of all we define functions, vectors, forms, coframe and frame. After that we define connection 1-forms. Finally we calculate curvature 2-forms, torsion 2-forms, Riemann and Ricci tensor fields.

restart:
with(atlas):

Function declaration:
Functions(f=f(x,y),h=h(x,y),g=g(x,y),z=z(x,y));

Vector fields:
Vectors(E[i],X,Y,Z);

Differential p-forms:
Forms(e[j]=1);

Coframe 1-forms:
Coframe(e[1]=d(x),e[2]=d(y));

Frame vector fields:
Frame(E[i]);

Connection definition:
omega[1,1]:=f*e[2];

omega[1,2]:=0;

omega[2,1]:=0;

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

Connection declaration:
Connection(omega);

Curvature calculation:
Curvature(Omega);

Result:
eval(Omega);

Torsion calculation:
Torsion(T);

Result:
eval(T);

Curvature tensor calculation:
Riemann(R);

Ricci tensor calculation:
Ricci(r);

Some more simple calculations:

Covariant derivatives:
'cov(E[j],e[1])'=cov(E[j],e[1]);
'cov(E[j],e[2])'=cov(E[j],e[2]);

Lie derivative:
'L[E[1]](E[2])'=L[E[1]](E[2]);

Interior products:
'iota[E[k]](T[1])'=iota[E[k]](T[1]);
'iota[E[k]](Omega[1,1])'=iota[E[k]](Omega[1,1]);