content_image/353613.jpg content_image/353612.jpg content_image/353611.jpg content_image/353610.jpg content_image/353608.jpg content_image/353609.jpg
## 2. THE CLASSICAL MODEL 

The characteristic feature of the Bianchi IX model is the existence of the simply transitive isometry group. The infinitesimal generators of this group are the three linearly independent spacelike killing vectors, ξ$_{i}$, which obey 

<img src='content_image/353613.jpg'>

where i , j and k runs from 1 to 3. The generators of these transformations are 

One may write the metric of Bianchi IX in terms of forms , 

<img src='content_image/353612.jpg'>

<img src='content_image/353611.jpg'>

where N is the lapse function, h$_{ij}$ is the 3-metric and the σ$_{i}$ are the left invariant 1-forms of SU (2): 

with 0 ≤ ψ ≤ 4 π , 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2 π . 

<img src='content_image/353610.jpg'>

Without lose of generality, we can assume that the metric h$_{ij}$ is diagonal. Using the Misner variables to parameterize the metric 

where α and β are functions of time only. The matrix 

<img src='content_image/353608.jpg'>

<img src='content_image/353609.jpg'>

with the property T rβ = 0, ensures that the 3-volume of the hypersurface depends only on the conformal factor α . The variables β$_{+}$ and β − describe the anisotropy of the spacetime, 

