Let $\xi$ be a real vector bundle over a base space $B$. It is my understanding a metric is meant as a function,
$$\beta : E(\xi \oplus \xi) \to \mathbb R$$
where $E$ denotes the total space of the Whitney sum, such that, $\beta$ restricted to $p^{-1}(b) \times p^{-1}(b)$ is an inner product on $p^{-1}(b)$ for all $b \in B$. ($p$ is the projection.)
I am also aware the existence of a Gauss map guarantees that a real vector bundle admits a metric. I thus have two questions:
- Is there no requirement that $\beta$ vary continuously from fibre to fibre?
- Apart from the explicit construction, is there an intuitive reason why the existence of a Gauss map would guarantee the existence of metric?
$\beta$ is required to be a continuous map.
I do not see how a Gauss map (see https://en.wikipedia.org/wiki/Gauss_map) should be related to a metric on a vector bundle. The standard approach is this: Locally any bundle is trivial, hence locally you can find a metric. Then a partition of unity is used to piece together the local metrics to a global metric. See for example Hatcher http://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf p.11.
Edited:
A Gauss map in the sense of this question is a continuous map $g : E(\xi) \to \mathbb{R}^m$ such that the restriction to each fiber is a linear monomorphism. Let $s : \mathbb{R}^m \times \mathbb{R}^m \to \mathbb{R}$ be the usual inner product. We obtain a continuous map $$\mu : E(\xi \oplus \xi) \to E(\xi) \times E(\xi) \stackrel{g \times g}{\rightarrow} \mathbb{R}^m \times \mathbb{R}^m \stackrel{s}{\rightarrow} \mathbb{R}$$ which is the desired metric.
By the way, a Gauss map can be regarded as an embedding $\hat{g}$ of $\xi$ into the trivial m-dimensional bundle $\tau_m$ over $B$ (having total space $B \times \mathbb{R}^m$). In fact, define $\hat{g}(e) = (p(e),g(e))$.