ds2=dt2+a(t)2(1)2+b(t)2(2)2+c(t)2(3)2,ds^2 = -dt^2 + a(t)^2 (\omega^1)^2 + b(t)^2 (\omega^2)^2 + c(t)^2 (\omega^3)^2,
where i\omega^i are left-invariant 1-forms on SU(2), satisfying:
di=12 jkijk.d\omega^i = \frac{1}{2} \epsilon^i_{\ jk} \omega^j \wedge \omega^k.
In terms of Euler angles (,,)(\theta, \phi, \psi), the 1-forms are explicitly:
1=cosd+sinsind,2=sindcossind,3=d+cosd.\begin{aligned} \omega^1 &= \cos\psi\, d\theta + \sin\psi\, \sin\theta\, d\phi, \\ \omega^2 &= \sin\psi\, d\theta - \cos\psi\, \sin\theta\, d\phi, \\ \omega^3 &= d\psi + \cos\theta\, d\phi. \end{aligned}
These forms encode the rotational symmetry of the 3-sphere, and the time-dependent scale factors a(t),b(t),c(t)a(t), b(t), c(t) evolve according to modified Einstein--Cartan dynamics.
The Christoffel symbols, Ricci tensor components, and curvature scalar can be computed from this metric using standard methods or symbolic computation tools (e.g., xAct in Mathematica). This leads to the anisotropic generalization of the Friedmann equations, with additional terms from shear and spatial curvature anisotropies.
Appendix B: Torsion Field Equations in Riemann--Cartan Geometry
In Riemann--Cartan spacetime, the affine connection includes antisymmetric parts:
~=+K,\tilde{\Gamma}^\lambda_{\;\mu\nu} = \Gamma^\lambda_{\;\mu\nu} + K^\lambda_{\;\mu\nu},
where KK^\lambda_{\;\mu\nu} is the contortion tensor, and the torsion tensor is:
