Differential form ''equation''
$begingroup$
I am having a bit of a trouble with the following. I'm working in the homogeneous Lie Group $mathbb{R}ltimes mathbb{R}^3$ with an specific bracket an it give me de following system to integrate and get the metric in coordinates:
$$
begin{array}{l}
de^1=e^1wedge e^4+beta e^2wedge e^4- gamma e^3wedge e^4\
de^4=-beta e^1wedge e^4+alpha e^2wedge e^4+delta e^3wedge e^4\
de^3=gamma e^1wedge e^4-delta e^2wedge e^4-dfrac{alpha}{alpha+1}e^3wedge e^4\
de^4=0,\
d^2e^i=0
end{array}
$$
where $e^i$ is de dual basis of $e_i$ and $alpha$ is a real number ann the other coefficients are combinations of it. The idea is put this $e^i$ in coordinates of $mathbb{R}^4, quad (x,y,z,t)$, to get the metric tensor in coordinates $$g=e^4otimes e^4+e^4otimes e^4+e^4otimes e^4+e^4otimes e^4$$
For example, $e^4=dt$, as it fulfil the conditions, I don't need a general solution, just a particular one. I hope you all get this properly, any help could be great (general ideas to solve it, some mathematica code...).
riemannian-geometry lie-algebras differential-forms
asked Jan 15 at 16:33
R4MR4M
12
$endgroup$
add a comment |
$begingroup$
I am having a bit of a trouble with the following. I'm working in the homogeneous Lie Group $mathbb{R}ltimes mathbb{R}^3$ with an specific bracket an it give me de following system to integrate and get the metric in coordinates:
$$
begin{array}{l}
de^1=e^1wedge e^4+beta e^2wedge e^4- gamma e^3wedge e^4\
de^4=-beta e^1wedge e^4+alpha e^2wedge e^4+delta e^3wedge e^4\
de^3=gamma e^1wedge e^4-delta e^2wedge e^4-dfrac{alpha}{alpha+1}e^3wedge e^4\
de^4=0,\
d^2e^i=0
end{array}
$$
where $e^i$ is de dual basis of $e_i$ and $alpha$ is a real number ann the other coefficients are combinations of it. The idea is put this $e^i$ in coordinates of $mathbb{R}^4, quad (x,y,z,t)$, to get the metric tensor in coordinates $$g=e^4otimes e^4+e^4otimes e^4+e^4otimes e^4+e^4otimes e^4$$
For example, $e^4=dt$, as it fulfil the conditions, I don't need a general solution, just a particular one. I hope you all get this properly, any help could be great (general ideas to solve it, some mathematica code...).
riemannian-geometry lie-algebras differential-forms
asked Jan 15 at 16:33
R4MR4M
12
$endgroup$
add a comment |
$begingroup$
I am having a bit of a trouble with the following. I'm working in the homogeneous Lie Group $mathbb{R}ltimes mathbb{R}^3$ with an specific bracket an it give me de following system to integrate and get the metric in coordinates:
$$
begin{array}{l}
de^1=e^1wedge e^4+beta e^2wedge e^4- gamma e^3wedge e^4\
de^4=-beta e^1wedge e^4+alpha e^2wedge e^4+delta e^3wedge e^4\
de^3=gamma e^1wedge e^4-delta e^2wedge e^4-dfrac{alpha}{alpha+1}e^3wedge e^4\
de^4=0,\
d^2e^i=0
end{array}
$$
where $e^i$ is de dual basis of $e_i$ and $alpha$ is a real number ann the other coefficients are combinations of it. The idea is put this $e^i$ in coordinates of $mathbb{R}^4, quad (x,y,z,t)$, to get the metric tensor in coordinates $$g=e^4otimes e^4+e^4otimes e^4+e^4otimes e^4+e^4otimes e^4$$
For example, $e^4=dt$, as it fulfil the conditions, I don't need a general solution, just a particular one. I hope you all get this properly, any help could be great (general ideas to solve it, some mathematica code...).
riemannian-geometry lie-algebras differential-forms
asked Jan 15 at 16:33
R4MR4M
12
$endgroup$
I am having a bit of a trouble with the following. I'm working in the homogeneous Lie Group $mathbb{R}ltimes mathbb{R}^3$ with an specific bracket an it give me de following system to integrate and get the metric in coordinates:
$$
begin{array}{l}
de^1=e^1wedge e^4+beta e^2wedge e^4- gamma e^3wedge e^4\
de^4=-beta e^1wedge e^4+alpha e^2wedge e^4+delta e^3wedge e^4\
de^3=gamma e^1wedge e^4-delta e^2wedge e^4-dfrac{alpha}{alpha+1}e^3wedge e^4\
de^4=0,\
d^2e^i=0
end{array}
$$
where $e^i$ is de dual basis of $e_i$ and $alpha$ is a real number ann the other coefficients are combinations of it. The idea is put this $e^i$ in coordinates of $mathbb{R}^4, quad (x,y,z,t)$, to get the metric tensor in coordinates $$g=e^4otimes e^4+e^4otimes e^4+e^4otimes e^4+e^4otimes e^4$$
For example, $e^4=dt$, as it fulfil the conditions, I don't need a general solution, just a particular one. I hope you all get this properly, any help could be great (general ideas to solve it, some mathematica code...).
riemannian-geometry lie-algebras differential-forms
riemannian-geometry lie-algebras differential-forms
asked Jan 15 at 16:33
R4MR4M
12
asked Jan 15 at 16:33
R4MR4M
12
asked Jan 15 at 16:33
R4MR4M
12
asked Jan 15 at 16:33
R4MR4M
12
asked Jan 15 at 16:33
R4MR4M
12
12
add a comment |
add a comment |
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