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:

http://en.wikipedia.org/wiki/Reduction_of_the_structure_group

http://www.math.washington.edu/~julia/Quillen/prin.pdf

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".

Thank you

equivalentto the existence of a metric: ams.org/journals/proc/1969-020-02/S0002-9939-1969-0236876-3/… – Otis Chodosh Jun 28 '13 at 22:00