Suppose I have two Euclidean metrics $\mu_1, \mu_2$ on a given vector bundle $\xi$. Does anyone know of necessary and/or sufficient conditions to ensure that there is a homeomorphism $\phi: E(\xi) \to E(\xi)$ between fibers such that $\mu_1 \circ \phi = \mu_2$?
I vaguely recall results relating to this question, but it's been a while since I studied algebraic topology.