Let $ E \longrightarrow B $ be a vector bundle.
I know that if B is paracompact, the bundle admits a metric.
My question is the following: is this true for any B?
I also know that a reduction of the structure group from $GL(n,R)$ to $O(n)$ is possible but I don't know about the conditions on B.
It seems to me that if it were true for any B, then any vector bundle would admit a metric. This seems strange
Here are two links that have caused confusion:
p.19 "In other words, every real vector bundle admits a Euclidean metric. Similarly, every GLnC-bundle is induced from a U(n)-bundle; equivalently, every complex vector bundle admits a Hermitian metric".