= Splitting a trivial line from a nowhere-zero section
{title2=$E\cong\sigma^\perp\oplus\mathbb R$}
A unit <nowhere-zero section> $\sigma$ of a real <vector bundle> with a <fiber metric> defines a rank-one <trivial vector bundle>. Its fiberwise <orthogonal complement> is a subbundle: locally complete $\sigma$ to an orthonormal frame. The <vector bundle isomorphism> $(v,t)\mapsto v+t\sigma$ has inverse $w\mapsto(w-\langle w,\sigma\rangle\sigma,\langle w,\sigma\rangle)$. For an oriented bundle, orient the complement so that its positive frame followed by $\sigma$ is positive.
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