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

TL;DR

This paper revisits diagonal Einstein metrics for Bianchi types II, III, and V, exploring their derivations, integrability of geodesic flow, and signature-dependent bifurcation phenomena involving elliptic and elementary functions.

Contribution

Provides simplified derivations of Einstein metrics for Bianchi types II, III, and V, highlighting integrability and bifurcation phenomena based on signature.

Findings

Geodesic flow is integrable for Bianchi types II and III.

Bifurcation phenomenon occurs in type V metrics depending on signature.

Elliptic functions are essential in Minkowskian type V metrics, elementary functions in Euclidean.

Abstract

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.