Conformally flat metric on 2-dimentional sphere 

Description: 

This worksheet illustrates how to use atlas  package to solve problems in elementary differential geometry. As an example we discuss conformally flat metric on 2-dimensional sphere 

 

We construct the metric on sphere and calculate connection 1-forms , curvature 2-forms , curvature tensor field (Riemann   tensor  field), Ricci tensor field and Ricci scalar expression which is proportional to scalar curvature . We also calculate some additional quantities: Lie derivatives, exterior derivatives, interior products etc. 

 

Shpere - : 

>	restart: with(atlas):

 

Conformally flat metric on sphere : 

Declare constant :
Constants(lambda); 

(2.1)

 

Declare functions:
Functions(f=f(x,y),h=h(f)); 

(2.2)

 

Declear forms:
Forms(e[i]=1,xi=1,theta=p); 

(2.3)

 

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

(2.4)

 

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

(2.5)

 

Now coframe 1-forms are . 

Declare frame vectors:
Frame(E[j]); 

(2.6)

 

Now frame vectors are . 

Metric declaration:
Metric(g=4*(d(x)&.d(x)+d(y)&.d(y))/(1+lambda*(x^2+y^2))^2); 

(2.7)

 

Calculate connection 1-forms:
Connection(omega); 

(2.8)

 

Calculate curvature 2-forms:
Curvature(Omega); 

(2.9)

 

Riemann tensor:
Riemann(R); 

(2.10)

 

Ricci tensor:
Ricci(r); 

(2.11)

 

Ricci scalar:
RicciScalar(s); 

(2.12)

 

Show 1-form :
'omega[2,1]'=factor(omega[2,1]); 

(2.13)

 

Verify that there is no Killing vector field among frame vector fields:
'L[E[j]](g)'=L[E[j]](g); 

(2.14)

 

Verify that "rotation" vector field  is a Killing one:
Z:=x*E[2]-y*E[1];
'L[Z]'(g)=L[Z](g); 

 

	(2.15)

 

Calculation of volume form using Hodge operator &** :
'&**(1)'=radsimp(&**(1)); 

(2.16)

 

Some more calculations: 

Using interior product operator - :
'iota[E[j]](&**(1))'=iota[E[j]](radsimp(&**(1))); 

(2.17)

 

The same with metric tensor g:
'iota[E[k]](g)'=iota[E[k]](g); 

(2.18)

 

Using exterior derivative operator - :
'd(f)'=d(f); 

(2.19)

 

>	'd(h)'=d(h);

 

(2.20)

 

>	'd(f*xi)'=d(f*xi);

 

(2.21)

 

>	'&**(d(f))'=radsimp(&**(d(f)));

 

(2.22)

 

As are coframe 1-forms and are frame vectors then:
'iota[E[i]](e[k])'=iota[E[i]](e[k]); 

(2.23)

 

"ToBasis" decomposition:
X=ToBasis(X);
xi=ToBasis(xi); 

 

	(2.24)

 

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