Definition:Smooth Fiber Metric

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Let $E \to M$ be a smooth vector bundle.

Let $p \in E$.

Let $E_p$ be a fiber at $p$.

Let $\sigma, \tau$ be smooth sections of $E$.

Suppose on each $E_p$ the inner product $\innerprod \sigma \tau$ is a smooth function.

Then the inner product $\innerprod \cdot \cdot$ is called the smooth fiber metric (on $E$).