Bianchi Classification - Structure Constants

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

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