= Orthogonal structure on a real vector bundle
{title2=$O(r)$}
= Orthogonal structure
{synonym}
An orthogonal structure is a reduction of the transition functions of a real rank-$r$ <vector bundle> to the <orthogonal group>. Its orthonormal frames define a <fiber metric>, independent of the chosen frame because orthogonal transitions preserve the <inner product>. Conversely a <fiber metric> gives smooth orthonormal frames by the <Gram-Schmidt process>, with orthogonal transition matrices. This equivalence does not require the <vector bundle> to be globally trivial.
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