Hermitian metric on a vector bundle

Remarks

In a local frame $e_1, \dots, e_r$ of $E$ over $U$, a Hermitian metric is the same as a smooth map $H = (h(e_j, e_i))_{i,j}$ from $U$ to the positive definite Hermitian $r \times r$ matrices. A vector bundle with a Hermitian metric is called a Hermitian vector bundle. When $X$ is a complex manifold and $E$ is its holomorphic tangent bundle, a Hermitian metric on $E$ is a Hermitian metric on $X$ in the sense of Hermitian manifolds.

Example: on $\mathcal{O}(-1) \subset \mathbb{P}^n \times \C^{n+1}$, restrict the standard inner product $\sum_k v_k \bar w_k$ of $\C^{n+1}$ to each fibre. Each restriction is a Hermitian inner product, and for a local holomorphic section such as $s(z) = (1, z_1, \dots, z_n)$ on $\set{z_0 \ne 0}$ one gets $h(s, s) = 1 + \abs{z_1}^2 + \cdots + \abs{z_n}^2$, which is smooth.

Non-example: on the trivial line bundle $\C \times \C \to \C$, the rule $h_z(v, w) = \abs{z}^2 v \bar w$ is smooth, but $h_0(v, v) = 0$ for all $v$, so clause 1 fails at $z = 0$.