Bianchi spaces and their 3-dimensional isometries as S-expansions of 2-dimensional isometries

TL;DR

This paper demonstrates that certain 3D Bianchi isometry algebras can be derived from 2D isometries using S-expansions, revealing which algebras are obtainable and highlighting intrinsic 3D properties.

Contribution

It shows that some 3D Bianchi algebras are obtainable as S-expansions of 2D isometries, and identifies which types can or cannot be derived this way.

Findings

Types I, II, III, V can be obtained from 2D isometries.

Types IV, VI-IX cannot be obtained from 2D isometries.

Multiple semigroups can lead to the same algebra.

Abstract

In this paper we show that some 3-dimensional isometry algebras, specifically those of type I, II, III and V (according Bianchi's classification), can be obtained as expansions of the isometries in 2 dimensions. It is shown that in general more than one semigroup will lead to the same result. It is impossible to obtain the algebras of type IV, VI-IX as an expansion from the isometry algebras in 2 dimensions. This means that the first set of algebras has properties that can be obtained from isometries in 2 dimensions while the second set has properties that are in some sense intrinsic in 3 dimensions. All the results are checked with computer programs. This procedure can be generalized to higher dimensions, which could be useful for diverse physical applications.