Bianchi type II,III and V diagonal Einstein metrics re-visited

We present, for both minkowskian and euclidean signatures, short derivations of the diagonal Einstein metrics for Bianchi type II, III and V. For the first two cases we show the integrability of the geodesic flow while for the third case a somewhat unusual bifurcation phenomenon takes place: for minkowskian signature elliptic functions are essential in the metric while for euclidean signature only elementary functions appear.

💡 Research Summary

The paper revisits the diagonal Einstein metrics for Bianchi types II, III and V, providing compact derivations for both Lorentzian (Minkowskian) and Euclidean signatures. Starting from the most general left‑invariant one‑forms σi (i=1,2,3) of the respective Bianchi groups, the authors assume a diagonal line element

 ds² = ε dt² + a(t)² σ₁² + b(t)² σ₂² + c(t)² σ₃²,

where ε = –1 for Lorentzian and ε = +1 for Euclidean signature. Substituting this ansatz into the Einstein equations Rμν = Λ gμν yields a set of ordinary differential equations for the three scale factors a(t), b(t) and c(t). Because the metric is diagonal, all off‑diagonal components of the Ricci tensor vanish automatically, leaving only the diagonal equations. A crucial first integral emerges in every case: the product a b c is constant up to a rescaling of the time coordinate. This reduces the dynamical system to two independent equations.

Bianchi type II (Heisenberg group).
The structure constants satisfy