Let $\pi: E \rightarrow B$ be a fibration over a Riemannian manifold $B$, with $\pi^{-1}[b]$ homeomorphic to $\mathbb{R}$.

More precisely:

- I want each fiber $\pi^{-1}[b]=Im(f_b)$ for some $C^{\infty}$-curve $f_b:\mathbb{R}\rightarrow \mathbb{R}$.
- If $(f_b(x),b),(f_b(y),b)$ lie on the same fiber (above the point $b\in B$) then: I would like the distance between them to be defined as:
\begin{equation}
d_b((f_b(x),b),(f_b(y),b)):=\frac{\int_x^y f_b(u) du}{f_b(1)=f_b(0)}.
\end{equation}
*(that is I want geodesics connecting two points on the curve $f_b$ above $b$ to be the corresponding segment of that curve. )* - I would like the distance between two points on the same "level" to but corresponding to different parameters $b,b'$'s distance to be measured purely in terms of B's distance, that is:
\begin{equation}
d((f_{b'}(x),b'),(f_b(x),b))=d_B(\pi(f_{b'}(x),b'),\pi(f_b(x),b))=d_B(b',b).
\end{equation}
*(that is I want geodesics connecting two points on different curves $f_b(x)$ and $f_{b'}(x)$ on the same level (x) to correspond to the distance between the parameters $b$ and $b'$ in $B$. )*

**Notes on Notation:**
Where $d_b$ is the Riemmanian metric existing on a specific fiber, $d_B$ is the Riemmanian Metric on $B$ and $d$ is the metric I am looking for on $E$.

**Question:**

Then can I make $E$ into a manifold and it with a Riemannian metric satisfying the above such that moreover:

each fiber $\pi^{-1}[b]$ is a geodesic in $E$?

**Nutshell Resume of Goal:**

(essentially I want to be able to move from $f_b(x)$ to $f_{b'}(y)$ in the shortest way given the above constraints, and explicitly calculate that distance)