connections2.html

> with(difforms):with(LinearAlgebra):with(combinat):

Warning, the protected name Chi has been redefined and unprotected

> Gamma:=proc(g::Matrix,coords::Vector)
local i,j,k,l,gi,gamm;
gi:=g^(-1);
gamm:=Array(1..RowDimension(g),1..RowDimension(g),1..RowDimension(g));
for i from 1 to RowDimension(g) do
for j from 1 to RowDimension(g) do
for k from 1 to RowDimension(g) do
gamm[i,j,k]:=(1/2)*add(gi[k,l]*(diff(g[j,l],coords[i])+diff(g[i,l],coords[j])-diff(g[i,j],coords[l])),l=1..RowDimension(g));
end do;
end do;
end do;
return(gamm);
end proc;

> g:=Matrix(2,2,[[1,0],[0,sin(phi1)^2]]);

> coords:=Vector([phi1,phi2]);

> G:=Gamma(g,coords);

> type(g,Matrix);

$true$

> RowDimension(g);

$2$

> g[1,1];

$1$

> g^(-1);

> G_(1,2,1);

$G_(1, 2, 1)$

> G[1,2,1];

$0$

> print(G);

> G[1];

> G[2];

> G[1,2,2];

$\frac{\cos (φ1)}{\sin (φ1)}$

> defform(phi1=scalar,phi2=scalar,phi3=scalar,phi4=scalar,phi5=scalar,e1=1,e2=1,e3=1,e4=1,e5=1);

> g:=Matrix(5,5,[[1,0,0,0,0],[0,sin(phi1)^2,0,0,0],[0,0,1,0,0],[0,0,0,sin(phi3)^2,0],[0,0,0,0,sin(phi3)^2*sin(phi4)^2]]);

> En:=Vector(5,symbol=E);

> en:=Vector(5,[e1,e2,e3,e4,e5]);

> Conn:=(i,j,b,G)->add(G[i,j,m]*b[m],m=1..Dimension(b));

> coords:=Vector(5,[phi1,phi2,phi3,phi4,phi5]);

> G:=Gamma(g,coords);

> Conn(1,1,En,G);

$0$

> CMatrix:=Matrix(5,5);

> for i from 1 to 5 do
for j from 1 to 5 do
CMatrix[i,j]:=Conn(i,j,En,G);
end do;
end do;

> CMatrix;

> CMatrix[1,2] mod En[2];

> mods(CMatrix[1,2],En[2]);

> CMatrix[1,2];

> coeffs(CMatrix[1,2],En[1]);

> coeff(CMatrix[1,2],En[5]);

$0$

> conforms:=proc(C::Matrix,B::Vector,DB::Vector)
local i,j,k,omega,n;
n:=RowDimension(C);
omega:=Matrix(n);
for i from 1 to n do
for j from 1 to n do
omega[i,j]:=add(coeff(C[k,i],En[j])*DB[k],k=1..n);
end do;
end do;
return(omega);
end proc;

> conforms(CMatrix,En,en);

>    RelativeParity:=proc(p1::list,p2::list)
description "determine the relative parity of two permutations";
  table( 'antisymmetric');
  op(1,%[p1[]])*op(1,%[p2[]])
end:

> Phi:=proc(k::integer,O::Matrix,o::Matrix)
local sum,temp,alpha,i,j,n;
n:=RowDimension(O);
sum:=0;
for alpha in permute(n-1) do
temp:=1;
for i from 1 to k do
temp := temp &^ O[alpha[2*i-1],alpha[2*i]];
end do;
for i from 2*k+1 to n-1 do
temp := temp &^ o[alpha[i],n];
end do;
sum:= sum + RelativeParity(alpha,[seq(i,i=1..n-1)])*temp;
end do;
end proc;

> CurvatureForm := (i,j,o) -> d(o[i,j]) - add(o[i,k] &^ o[k,j],k=1..RowDimension(o));

> PI:=proc(O::Matrix,o::Matrix)
local lambda,p,total,n;
n:=RowDimension(O);
total:=0;
if ((n mod 2) = 1) then
p := (n-1)/2;
for lambda from 0 to p do
total:=total+(-1)^lambda * binomial(p,lambda) * Phi(lambda,O,o);
end do;
return(1/(2^n * Pi^p * p!) * total);
else
p:=n/2;
for lambda from 0 to p-1 do
total:=total+(-1)^lambda * product(2*i-1,i=1..p-lambda)*Phi(lambda,O,o);
end do;
return 1/Pi^p * total;
end if;
end proc;

> CM2:=Matrix(3,[[0,(2*lambda-1/lambda)*En[3],(1/lambda-2*lambda)*En[2]],[-1/lambda*En[3],0,(1/lambda)*En[1]],[1/lambda*En[2],(-1/lambda)*En[1],0]]);

> omega:=conforms(CM2,En,en);

> omega:=conforms(CMatrix,En,en);

> Omega:=Matrix(5):

> for i from 1 to 5 do
for j from 1 to 5 do
Omega[i,j]:=CurvatureForm(i,j,omega);
end do;
end do;

> PI5:=PI(Omega,omega):

> simpform(simplify(simpform(PI5)));

>