 Orthonormal frame bundle

For a metric of signature $(r,s)$, the orthonormal frame bundle consists of all linear isometries $u:{\mathbb{R}}^{r,s}\to T_{x}M$. Right composition by $O(r,s)$ makes it a principal bundle. Its standard associated vector bundle is the tangent bundle, via $[u,v]\mapsto u(v)$. A global orthonormal frame is a global section and therefore trivializes this principal bundle. A Lie group with a left-invariant metric has such a frame by left translating an orthonormal basis at the identity.