Riemannian Geometry/Bundle metrics

Definition (bundle metric):

Let $E\to M$ be a smooth vectur bundle. A bundle metric on $M$ is a family $\langle \cdot ,\cdot \rangle _{p}$ ($p\in M$), where each $\langle \cdot ,\cdot \rangle _{p}$ is a scalar product on $E_{p}$, which satisfies the following smoothness condition: For all $\sigma ,\tau \in \Gamma (E)$, the map

$p\mapsto \langle \sigma (p),\tau (p)\rangle _{p}$

is smooth, ie. contained in ${\mathcal {C}}^{\infty }(M)$.

Definition (Riemannian metric):

Let $M$ be a smooth manifold. A Riemannian metric is a bundle metric on the tangent space $TM$.