= Horizontal connection associated to a covariant derivative
{title2=$H_{(x,v)}=\{(\xi,-\Gamma_x(\xi)v)\}$}
In a <frame of a vector bundle>, write a <vector bundle covariant derivative> as $\nabla(e v)=e(dv+\Gamma v)$. The associated horizontal subspace at $(x,v)$ consists of tangent pairs $(\xi,-\Gamma_x(\xi)v)$. Under a frame change $e'=eg$, the <connection one-form> changes by $\Gamma'=g^{-1}\Gamma g+g^{-1}dg$, and this horizontal equation changes to the corresponding one in the new frame. Thus the subspaces glue to a unique smooth linear <connection on a vector bundle>. Its vertical projection gives back the original derivative. The <Leibniz rule> ensures locality, allowing this construction from an operator initially defined on global sections.
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