Let $0 \to E_1 \to E_2 \xrightarrow{\beta} E_3 \to 0$ be a short exact sequence of smooth vector bundles over a manifold $M$. After giving a metric to $E_2$ ($g_2$) for example using partitions of unity I want to show that I can equip $E_3$ with a metric satisfying,
$$g_3(u,u)=\operatorname{inf}\{ g_2(v,v) |\text{ with } \beta(v)=u \} $$
Now I have two questions: Is this naive definition of the metric correct? $$g_3(u_1,u_2)=\operatorname{inf}\{ g_2(v_1,v_2) |\text{ with } \beta(v_i)=u_i \} $$
Is this infimum actually a minimum for the case $u_1=u_2$?
The naive definition is correct, but things can be done in a simpler way. Exactness easily implies that for a point $x\in M$, the map $\beta_x$ restricts to a linear isomorphism between $ker(\beta_x)^\perp$ and the fiber of $E_3$ over $x$. Now you simply carry over the restriction of $g_2$ to this subbundle to $E_3$ via this isomorphism. In particular, this shows that the infimum is always attained (and where it is attained).