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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