Suppose that I'm given a Riemmanian manifold $<B,g_B>$ and a real line bundle $\pi:E\rightarrow B$, such that each fiber above any $b\in B$ comes equipped with a Riemmanian metric $g_b$.
Then how can I construct a Riemmanian metric $G$ on $E$, such that $\pi$ is a Riemmanian Submersion under this metric $G$?
Do I need a choice of connection> If so how do I make that choice?
(I'm quite new to this subject)
$\newcommand{\Reals}{\mathbf{R}}$If $B$ is $n$-dimensional, then locally $TB$ is modeled by $TU \simeq U \times \Reals^{n} \to U$, the line bundle $E$ is modeled by $\pi:U \times \Reals \to U$, and $TE$ is locally modeled by $$ T(U \times \Reals) \simeq \pi^{*}(TU) \oplus \pi^{*}(U \times \Reals). $$
Globally, the tangent bundle of the total space $E$ may be identified with $$ \pi^{*} (TB \oplus E) \simeq \pi^{*}(TB) \oplus \pi^{*} E. \tag{1} $$ The second summand is $\ker(d\pi) \subset TE$, but the first summand comes from a quotient, so a choice is required to identify $\pi^{*}(TB)$ as a subbunde of $TE$. Any particular identification defines an Ehresmann connection. The respective summands $\pi^{*}(TB) \to E$ and $\pi^{*}E \to E$ are the so-called horizontal and vertical subbundles.
Once an identification (1) is fixed, every vector $v$ tangent to some point of $\pi^{-1}(b) \subset E$ decomposes uniquely into horizontal and vertical components: $v = v_{H} + v_{V}$, with $v_{V} \in E_{b}$, the fibre of $E$ obver $b$, and (modulo identification) $v_{H} \in T_{b}B$.
In terms of the base metric $g_{B}$ on $B$ and the fibre metric $g$ on $E$, one can define $$ G(v, v) = g_{B}(v_{H}, v_{H}) + g(v_{V}, v_{V}). $$ The projection $\pi:(E, G) \to (B, g_{B})$ is a Riemannian submersion.