Conformally flat metric on 2-dimensional sphere

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Conformally flat metric on 2-dimentional sphere

Copyright © 2004-2008 DigiArea Group . All rights reserved.

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:

simple1

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 function which is proportional to scalar curvature . We also calculate some additional quantities: Lie derivatives, exterior derivatives, interior products etc.

Load atlas package:

restart:
with(atlas):

Conformally flat metric on sphere :

Declare constant :
Constants(lambda);

${Catalan, _Z, Pi, I, lambda, -I}$

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

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

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

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

Declare frame vectors:
Frame(E[j]);

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

g = 4*(`&.`(e[1],e[1])+`&.`(e[2],e[2]))/(1+lambda*(x^2+y^2))^2

Calculate connection 1-forms:
Connection(omega);

Let us see the result:
eval(omega);

TABLE([(1, 2) = -2/(1+lambda*x^2+lambda*y^2)*lambda*y*e[1]+2*lambda*x/(1+lambda*x^2+lambda*y^2)*e[2], (1, 1) = -2*lambda*x/(1+lambda*x^2+lambda*y^2)*e[1]-2/(1+lambda*x^2+lambda*y^2)*lambda*y*e[2], (2, ...

Calculate curvature 2-forms:
Curvature(Omega);

Let us see the result:
eval(Omega);

TABLE([(1, 2) = 4*lambda/(1+lambda*x^2+lambda*y^2)^2*`&^`(e[1],e[2]), (1, 1) = 0, (2, 1) = -4*lambda/(1+lambda*x^2+lambda*y^2)^2*`&^`(e[1],e[2]), (2, 2) = 0])

Riemann tensor:
Riemann(R);

R = 16/(1+lambda*x^2+lambda*y^2)^4*lambda*`&.`(`&^`(e[1],e[2]),`&^`(e[1],e[2]))

Ricci tensor:
Ricci(r);

r = 4*lambda/(1+lambda*x^2+lambda*y^2)^2*`&.`(e[1],e[1])+4*lambda/(1+lambda*x^2+lambda*y^2)^2*`&.`(e[2],e[2])

Ricci scalar:
RicciScalar(s);

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

omega[2,1] = -2*lambda*(-y*e[1]+x*e[2])/(1+lambda*x^2+lambda*y^2)

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

L[E[j]](g) = -8*(`&.`(e[1],e[1])+`&.`(e[2],e[2]))*lambda/(1+lambda*x^2+lambda*y^2)^3*(2*x*delta[1,j]+2*y*delta[2,j])

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

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

`&**`(1) = 4/abs(1+lambda*x^2+lambda*y^2)^2*`&^`(e[1],e[2])

Some more calculations:

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

iota[E[j]](`&**`(1)) = 4/abs(1+lambda*x^2+lambda*y^2)^2*(delta[1,j]*e[2]-delta[2,j]*e[1])

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

iota[E[k]](g) = 4/(1+lambda*x^2+lambda*y^2)^2*(delta[1,k]*e[1]+delta[2,k]*e[2])

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

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

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

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

`&**`(d(f)) = (1+lambda*x^2+lambda*y^2)^2*(Diff(f,x)*e[2]-Diff(f,y)*e[1])/abs(1+lambda*x^2+lambda*y^2)^2

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

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