Definition of fiber-wise inner product on a complex vector bundle

Question

Let $M$ be a smooth ($C^{\infty}$) manifold. Let $\pi:E\to M$ be a complex vector bundle. Let $\Gamma(E)$ denote the $C^{\infty}$ sections of $\pi:E\to M$, i.e. mappings $s:M\to E$ such that $s$ is smooth as a map of manifolds and $\pi\circ s=\text{id}_{M}$.

This document talks about a "fiber-wise inner product on a (complex) vector bundle" (see page 5, definition 3.3). I have looked for a precise definition in different books such as "An Introduction to Manifolds" (L. W. Tu) and "Differential Forms and Connections" (R. W. R. Darling), without success. The latter does define inner products and metrics but in the (I think, restrictive) context of riemannian metrics.

As far as I understand, such an (hermitian) fiber-wise inner product would be defined in the following way.

An hermitian inner product on a complex vector bundle $\pi:E\to M$ is a mapping

$$\langle\cdot,\cdot\rangle:\Gamma(E)\times\Gamma(E)\to C^{\infty}(M,\mathbb{C})$$

defined as follows. Let $p\in M$ and let $s,s'\in\Gamma(E)$.

1. $\langle s,s'\rangle(p):=\langle s(p),s'(p)\rangle_{p}$
2. $\langle \cdot,\cdot\rangle_{p}$ is an hermitian inner product on the complex vector space $E_{p}:=\pi^{-1}(\{p\})$ (i.e. is an hermitian inner product on each fiber).

This seems to be the natural way to define an inner product on a vector bundle, but I don't know if this is well-defined.

Could you please give me reference works on the subject and/or "the right" definition of fiber-wise inner product, as intended by the author of the previously linked document?

EDIT: this document (see pages 15 & 16) seems to confirm that the previous definition is the correct idea. However, he seems to define the metric directly on each fiber. In other words:

An hermitian inner product on a complex vector bundle $\pi:E\to M$ is the data of an hermitian inner product $\langle \cdot,\cdot\rangle_{p}$ on each fiber $E_{p}:=\pi^{-1}(\{p\})$ such that the map $\langle s,s' \rangle:M\to\Bbb C:p\mapsto \langle s(p),s'(p)\rangle_{p}$ is smooth for any sections $s,s'\in\Gamma(E)$.

Answer 1

The idea of your definition is correct but like you noted there is a technical issue. An Hermitian inner product should be a "smoothly varying family of Hermitian inner products on each fiber". You suggest defining a Hermitian inner product as a function $\left< \cdot, \cdot \right> \colon \Gamma(E) \times \Gamma(E') \rightarrow C^{\infty}(M,\mathbb{C})$ on sections but then, without imposing additional conditions on $\left< \cdot, \cdot \right>$, there is no reason that such a function will "come from" functions $\left< \cdot, \cdot \right>_p \colon E_p \times E_p \rightarrow \mathbb{C}$ on each fiber. In other words, given such $\left< \cdot, \cdot \right>$, the expression $\left< \cdot, \cdot \right>_p$ has no meaning.

This can be solved in several equivalent ways:

1. You can say that a Hermitian inner product on $E$ is a family $\left< \cdot, \cdot \right>_p \colon E_p \times E_p \rightarrow \mathbb{C}$ of Hermitian inner products on each fiber $E_p$ such that if $s,s' \in \Gamma(E)$ are smooth sections of $E$ then the function $\left< s, s' \right> \colon M \rightarrow \mathbb{C}$ defined by $\left< s, s' \right>(p) := \left< s(p), s'(p) \right>_p$ is a smooth function. This reverses the logical order by first specifying how you want to act on each fiber and then requiring that the induced map on sections should map smooth sections to smooth functions.
2. You can say that a Hermitian inner product on $E$ is a map $\left< \cdot, \cdot \right> \colon \Gamma(E) \times \Gamma(E) \rightarrow C^{\infty}(M,\mathbb{C})$ that is $C^{\infty}(M,\mathbb{C})$-linear in the first variable and $C^{\infty}(M,\mathbb{C})$-conjugate linear in the second variable. A standard result in differential geometry shows that such a map is tensorial and comes from a family of maps $\left< \cdot, \cdot \right>_p \colon E_p \times E_p \rightarrow \mathbb{C}$ on each fiber and then you can impose the requirement that the maps $\left< \cdot, \cdot \right>_p$ are Hermitian inner products.
3. You can say that a Hermitian inner product on $E$ is a smooth section of the bundle $(E \otimes \overline{E})^{*}$ whose value at each point $p \in M$ is a Hermitian inner product on $E_p$.