Orthogonal structure on a real vector bundle ( $O (r)$ )

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.

Table of contents
- Euclidean orthonormal frame Orthogonal structure on a real vector bundle
- Orthogonal local trivialization Orthogonal structure on a real vector bundle

Euclidean orthonormal frame ( $g (e_{i}, e_{j}) = δ_{ij}$ )

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This is a frame of a real vector bundle with a positive-definite fiber metric for which the basis vectors are orthonormal. The Gram-Schmidt process applied to any smooth local frame produces such frames smoothly. They are distinguished from Lorentzian orthonormal frames, whose metric has mixed signature.

Orthogonal local trivialization ( $h (e_{i}, e_{j}) = δ_{ij}$ )

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A vector bundle trivialization is orthogonal when its coordinate frame is orthonormal for the fiber metric, so each fiber is identified isometrically with Euclidean space. Overlap changes then take values in the orthogonal group. Local orthogonal trivializations exist even when no global frame does.