In general relativity the Kantowski-Sachs metric (named after Ronald Kantowski and Rainer K. Sachs) describes a homogeneous but anisotropic universe whose spatial section has the topology of R × S 2 {\displaystyle \mathbb {R} \times S^{2}} . The metric is:
d s 2 = − d t 2 + e 2 Λ t d z 2 + 1 Λ ( d θ 2 + sin 2 θ d ϕ 2 ) {\displaystyle ds^{2}=-dt^{2}+e^{2{\sqrt {\Lambda }}t}dz^{2}+{\frac {1}{\Lambda }}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})}
The isometry group of this spacetime is R × S O ( 3 ) {\displaystyle \mathbb {R} \times SO(3)} . Remarkably, the isometry group does not act simply transitively on spacetime, nor does it possess a subgroup with simple transitive action.
See also Bianchi classification Dust solution
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