= Fiber metric
{title2=$h_x:E_x\times E_x\to\mathbb R$}
= Fibre metric
{synonym}
A fiber metric on a real smooth <vector bundle> is a positive-definite <inner product> on each fiber that varies smoothly in <vector bundle trivializations>. A smooth <partition of unity> subordinate to trivializing charts averages their Euclidean inner products. Nonnegative weights preserve positivity and local finiteness ensures smoothness. Hence every real vector bundle over a Hausdorff second-countable <smooth manifold> admits a fiber metric. Applied to the <tangent bundle>, this is existence of a <Riemannian metric>.
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