Structure Constants
The three-dimensional Bianchi spaces each admit a set of three Killing vectors which obey the following property:
where, the "structure constants" of the group, form a constant order-three tensor antisymmetric in its lower two indices. For any three-dimensional Bianchi space, is given by the relationship
where is the Levi-Civita symbol, is the Kronecker delta, and the vector and diagonal tensor are described by the following table, where gives the ith eigenvalue of ; the parameter a runs over all positive real numbers:
| Bianchi type | notes | ||||
|---|---|---|---|---|---|
| I | 0 | 0 | 0 | 0 | describes Euclidean space |
| II | 0 | 1 | 0 | 0 | |
| III | 1 | 0 | 1 | -1 | the subcase of type VIa with |
| IV | 1 | 0 | 0 | 1 | |
| V | 1 | 0 | 0 | 0 | has a hyper-pseudosphere as a special case |
| VI0 | 0 | 1 | -1 | 0 | |
| VIa | 0 | 1 | -1 | when, equivalent to type III | |
| VII0 | 0 | 1 | 1 | 0 | has Euclidean space as a special case |
| VIIa | 0 | 1 | 1 | has a hyper-pseudosphere as a special case | |
| VIII | 0 | 1 | 1 | -1 | |
| IX | 0 | 1 | 1 | 1 | has a hypersphere as a special case |
Read more about this topic: Bianchi Classification
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