this paper we assume the curvature assumption and write $\alpha = a/b$. There is no loss in generality in assuming that the pinching constants in satisfy $a \leq 1 \leq b$, for the ratio of maximum to minimum sectional curvature, $a/b$, is invariant under a homothety of the metric. Further, we follow the curvature sign conventions given in [@Lee] or [@Petersen]: for orthonormal vectors $X,Z$, the sectional curvature of the plane they span is $ \sec(X,Z) = R(X,Z,Z,X)$, where $R$ is the Riemannian curvature 4-tensor.

We trivialize the normal ray bundle with respect to the outward unit normal for $Y$ as $N^{+}Y \approx Y \times [0, \infty)$. The exponential map restricted to $N^{+}Y$ is written ${E}$. For $p \in Y$, the notation $\gamma_p$ denotes the geodesic normal to $Y$ emanating outwards from $p$. We call a geodesic ray *untrapped* if it eventually escapes any compact set.

Following Petersen [@Petersen] we reserve the term *geodesic segment* for a distance minimizing geodesic curve between two points.

It is easy to verify that $E\colon Y \times [0, \infty) {\rightarrow}M$ is a local diffeomorphism at every point of $Y \times \{0\}$. Therefore by compactness of $Y$ we may obtain an $\epsilon > 0$ and 