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

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.

23 pages, one of the authors is new, title corrected, finite semigroup programming is added, the semigroup construction procedure is checked by computer programs, references to semigroup programming are added, last section is extended, appendix added, discussion of all the types of Bianchi spaces is included