It is well-known that a topological group can be represented as a group of
isometries of a reflexive Banach space if and only if its topology is induced
by weakly almost periodic functions (see
\cite{Shtern:CompactSemitopologicalSemigroups},
\cite{Megrelishvili:OperatorTopologies} and
\cite{Megrelishvili:TopologicalTransformations}). We show that for a metrisable
group this is equivalent to the property that its metric is uniformly
equivalent to a stable metric in the sense of Krivine and Maurey (see
\cite{Krivine-Maurey:EspacesDeBanachStables}).