Connection 1-forms

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atlasWizard - Maplet™

atlas[Connection] - calculation of connection 1-forms

Calling Sequence:

Connection(Id)

Parameters:

Id - variable - connection identifier

Description:

The Connection procedure calculates or defines connection 1-forms. If a metric is defined then connection is calculated with respect to the metric otherwise just connection identifier is declared as a connection. If the identifier is a table then the table is declared as a connection (see example 3). To get the result of the calculation use eval or atlas[iota] operators.

Example 1

Abstract connection
restart:
with(atlas):

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

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

Declare coframe with 1-forms: (see atlas[Coframe] )
Coframe(e[k],k=1..n);

Declare frame vectors (see atlas[Frame] ):
Frame(E[i]);

Declaration of connection 1-forms:
Connection(omega);

Torsion(T);

'd(e[j])'=d(e[j]);

d(e[j]) = -Sum(`&^`(omega[j,l[1]],e[l[1]]),l[1] = 1 .. n)+T[j]

d(xi)=d(ToBasis(xi));

d(xi) = -Sum(`&^`(e[l[1]],d(iota[E[l[1]]](xi))),l[1] = 1 .. n)+Sum(iota[E[l[1]]](xi)*(-Sum(`&^`(omega[l[1],l[2]],e[l[2]]),l[2] = 1 .. n)+T[l[1]]),l[1] = 1 .. n)

Example 2

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);

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

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)));

[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]);

[E[1] = (1/2+1/2*lambda*x^2+1/2*lambda*y^2+1/2*lambda*z^2)*Diff(``,x), E[2] = (1/2+1/2*lambda*x^2+1/2*lambda*y^2+1/2*lambda*z^2)*Diff(``,y), E[3] = (1/2+1/2*lambda*x^2+1/2*lambda*y^2+1/2*lambda*z^2)*Di...

Basis one forms are not exact:
'd(e[1])'=normal(d(e[1]));

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);

Now connection 1-forms are calculated:
eval(omega);

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

Example 3

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]);

[E[1] = 1/(y^2+x^2)*x*Diff(``,x)+1/(y^2+x^2)*y*Diff(``,y), E[2] = -1/(y^2+x^2)*y*Diff(``,x)+1/(y^2+x^2)*x*Diff(``,y)]

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);

TABLE([(1, 2) = (-x+y^4+y^2*x^2)/(y^2+x^2)*`&^`(e[1],e[2]), (2, 2) = y*(3+x*y^2+x^3)/(y^2+x^2)*`&^`(e[1],e[2]), (2, 1) = x*(-3+x*y^2+x^3)/(y^2+x^2)*`&^`(e[1],e[2]), (1, 1) = -y*(-1+x*y^2+x^3)/(y^2+x^2)...

Torsion calculation:
Torsion(T);

eval(T);

TABLE([1 = y*`&^`(e[1],e[2]), 2 = (2+x*y^2+x^3)/(y^2+x^2)*`&^`(e[1],e[2])])

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