= Linear horizontal distribution on a vector bundle
{title2=$H_{(b,v)}=\{(X,-A_b(X)v):X\in T_bB\}$}
In a local trivialization, a vector-bundle connection with matrix $A$ defines the displayed complement of the vertical tangent space. The frame-change law makes these complements independent of the trivialization. Their dependence on the fiber coordinate is linear; an arbitrary horizontal complement need not satisfy this condition and need not define a <connection on a vector bundle>. Horizontal lifts satisfy the ordinary differential equation defining <parallel transport>.
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