Oriented real bundles and oriented frame bundles

For a rank-$n$ real vector bundle $\xi \to X$, form the orientation cover $\mathrm{Or}(\xi )\to X$ whose fiber is the set of orientations of ${\xi }_x$. In a linear chart, an orientation is the standard orientation or its negative, and a transition matrix acts by the sign of its determinant. These charts give a two-sheeted cover when $n\ge 1$.

An orientation of $\xi$ is a section of this cover, equivalently a continuous fiberwise choice of orientation. The zero vector space has its canonical orientation, so a rank-zero bundle has one orientation rather than two. A fiberwise invertible bundle map between oriented bundles is orientation-preserving when it carries the selected orientation to the selected orientation; in oriented local frames its matrices have positive determinant.

The oriented frame bundle ${\mathrm{Fr}}^{+}(\xi )\subseteq \mathrm{Fr}(\xi )$ consists of frames that transport the standard orientation of ${\mathbb{R}}^n$ to the selected orientation of ${\xi }_x$. Oriented linear charts identify it with $U\times {\mathrm{GL}}_n^{+}(\mathbb{R})$, and precomposition makes it a principal ${\mathrm{GL}}_n^{+}(\mathbb{R})$-bundle. This construction uses neither a metric nor a choice principle; a metric-dependent $\mathrm{SO}(n)$ reduction is treated separately.

For a subgroup $H\subseteq G$, an $H$-reduction of a principal $G$-bundle $P$ is a principal $H$-subbundle $Q\subseteq P$ for which $Q{\times }_{H}G\to P$, $[q,g]\mapsto qg$, is an isomorphism of the principal bundles defined in Principal g bundle and associated fiber bundle.