${\overline{E}}$ is a bijection. Relative to the diffeomorphism $E\colon Y \times [0, \infty) {\rightarrow}{M \backslash K}$ of the previous section, we write a generic curve $\sigma$ in ${M \backslash K}$ as $\sigma = (\sigma_Y, \sigma_r)$.

${\overline{E}}$ is injective. \[prop:Ebaroneone\]

We must show that given distinct points $p, q \in Y$ that the normal geodesics $\gamma_p, \gamma_q$ have unbounded distance as a function of time. It suffices to show given any $t > 0$ that the length of any curve $\sigma$ from $\gamma_p(t)$ to $\gamma_q(t)$ is bounded below by an unbounded function of time.

Suppose that $\sigma$ leaves the collar $Y \times [t/2, \infty)$. Then the normal contribution of the length integral and the decomposition of the metric imply $len(\sigma) \geq t$. In case that $\sigma$ remains in the collar, $len(\sigma) \geq len(\sigma^P)$, where $\sigma^P$ is the projection of $\sigma$ onto $Y \times \{t/2\}$, i.e. $\sigma^P(s) = (\sigma_Y(s), t/2)$. Then Jacobi field estimates imply that $$len(\sigma) \geq len( \sigma^P ) \geq \cosh(a t/2) d_Y(p,q).$$ Therefore the length of any curve from $\gamma_p(t)$ to $\gamma_q(t)$ is bounded below by an unbounded function of time.

In order to show that ${\overline{E}}$ is surjective we consider an untrapped geodesic ray $\sigma$ parametrized by arc 