= Isometry dimensions in dimension three
{title2=$0,1,2,3,4,6$}
For a connected three-dimensional <Riemannian manifold>, the <Myers-Steenrod theorem> bounds the dimension of its isometry group by six. Dimension five cannot occur: isotropy embeds in $O(3)$, whose Lie subalgebras have dimensions zero, one or three. A five-dimensional action would require three-dimensional isotropy and a two-dimensional orbit; full rotational isotropy has no invariant tangent two-plane. The maximal dimension six forces transitivity and isotropy, hence constant sectional curvature. Each of the six listed dimensions is attained by a suitable metric.
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