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A G-structure on a real rank-$m$ vector bundle is a principal $G$ reduction of its frame bundle. Equivalently it has local frames whose transition matrices take values in $G$, with the reduction consisting of their right $G$-orbits. An orthogonal structure on a real vector bundle is equivalent to a fiber metric; a special orthogonal reduction is equivalent to that metric together with an orientation of a vector bundle.

Take a locally finite trivializing cover $(U_{\alpha })$ of the real vector bundle $E$, and a subordinate smooth partition of unity $({\rho }_{\alpha })$. Each trivialization identifies its fibers with ${\mathbb{R}}^m$ and hence supplies a smooth Euclidean inner product $g_{\alpha }$ there. DefineEach weighted term extends by zero outside its chart, since the weight has support within that chart. Local finiteness makes the sum smooth in every bundle chart. At each $b$ at least one weight is positive; for $v\ne 0$ every nonzero contribution to $g_b(v,v)$ is positive. Thus $g$ is a smooth positive-definite fiber metric, which is precisely an inner product varying smoothly with the fibers.

A G-structure on a vector bundle is a reduction of its frame bundle from $GL(m,\mathbb{R})$ to $G$: a principal $G$-subbundle $P\subseteq \mathrm{Fr}(E)$ whose extension of structure group is the full frame bundle. Equivalently, it is a choice of smooth local frames with transition matrices in $G$, and the reduction consists of the right $G$-orbits of these frames. The adapted frames are part of the structure, not merely a statement that an individual fiber has a basis.

Given a fiber metric, the Gram-Schmidt process transforms any smooth local frame into Euclidean orthonormal frames. At each step the squared norm to be divided out is positive, so the divisions and positive square roots depend smoothly on the base point. Two such frames differ by an orthogonal matrix. All the orthonormal frames therefore form a principal $O(m)$ reduction, an orthogonal structure on a real vector bundle.

Conversely, let $P$ be an $O(m)$ reduction. A frame $u:{\mathbb{R}}^m\to E_b$ in $P_b$ definesReplacing $u$ by $uq$ for $q\in O(m)$ leaves this rule unchanged, since $q$ preserves the Euclidean inner product. It is consequently independent of the adapted frame and defines a positive-definite fiber metric. Its coefficients are smooth in local adapted frames. These constructions are mutually inverse. An $O(m)$-structure is equivalent to a smooth fiberwise inner product.

For $SO(m)$, the additional condition is an orientation of a vector bundle. Such a reduction supplies consistently oriented orthonormal frames, since its transition determinants are $+1$. Conversely an orientation together with a fiber metric selects the oriented Euclidean orthonormal frames and gives an $SO(m)$ reduction. ThusIt is not true for every real bundle. The Möbius line bundle is obtained from $[0,2\pi ]\times \mathbb{R}$ by $(2\pi ,v)\sim (0,-v)$. If it had an orientation, hence a nowhere-zero continuous section, its coefficient on the interval would satisfy $f(2\pi )=-f(0)$ while staying nonzero. The intermediate value theorem makes that impossible. Since $SO(1)=\{1\}$, it has no $SO(1)$ reduction. For each $m\ge 1$, its direct sum with a trivial rank-$(m-1)$ bundle has transition matrix $\mathrm{diag}(-1,1,\dots ,1)$ and determinant line equal to the Möbius line. It is also nonorientable and has no $SO(m)$-structure. This obstruction concerns the bundle's orientation, independently of the existence of a metric.

An orthogonal structure on a real vector bundle $E$ of rank $r$ is a choice of local frames whose transition matrices lie in the orthogonal group $O(r)$. In such frames declareOrthogonal transitions preserve this expression, so it defines a smooth positive-definite fiber metric. Conversely, a fiber metric turns any local frame into a smooth orthonormal frame by the Gram-Schmidt process; the resulting transition matrices are orthogonal. The two constructions are inverse, giving the required equivalence.

An orthogonal local trivialization $E{|}_U\cong U\times {\mathbb{R}}^r$ identifies each fiber isometrically with standard Euclidean space. Equivalently, its coordinate frame satisfies $h(e_i,e_j)={\delta }_{ij}$. This is local and does not assert the existence of a global frame.

Every real vector bundle over a smooth manifold admits a fiber metric. Choose a trivializing open cover and local Euclidean metrics $h_i$. The partition of unity theorem for a Hausdorff second-countable smooth manifold supplies nonnegative smooth functions ${\rho }_i$, with locally finite supports contained in the respective cover members, such that ${\sum }_i{\rho }_i=1$. Setextending each weighted metric by zero off its chart. Local finiteness makes this smooth. For every nonzero fiber vector $v$, at least one positive weight gives $h(v,v)={\sum }_i{\rho }_i{h}_i(v,v)>0$. Thus $h$ is a fiber metric and provides an orthogonal structure.

One definition of an orthogonal connection is that its covariant derivative preserves the fiber metric:for all vector fields $X$ and smooth sections of a vector bundle $s,t$. Another is that its parallel transport along every smooth curve is an isometry between the endpoint fibers.

For the first implication, take parallel sections $s,t$ along a curve. Their covariant derivatives vanish, so the displayed identity makes $h(s,t)$ constant: parallel transport preserves a fibre metric. Conversely, if parallel transport preserves the inner product, choose parallel extensions of arbitrary initial fiber vectors along a curve with prescribed initial tangent $X$. Differentiating their constant inner product gives $({\nabla }_{X}h)(s,t)=0$ at the initial point. Since $X,s,t$ were arbitrary, the connection is metric-compatible.

The equivalent local description is that the connection matrix in an orthonormal frame is skew-symmetric. Writing $\nabla e_j={\sum }_i{A}^i{}_j{e}_i$ in an orthogonal trivialization, differentiate $h(e_i,e_j)={\delta }_{ij}$ to obtainConversely this matrix identity gives the metric-preservation formula by expanding arbitrary sections. It is the orthogonal group connection description of the same condition.