normal ray bundle of $Y$. The main result of this paper is:

Let $(M,g)$ be a complete, noncompact Riemannian $(n+1)$-manifold. Suppose that there exists $K \subset M$, a compact $(n+1)$-dimensional Riemannian submanifold with boundary that satisfies:

1.  $K$ is totally convex in $M$, i.e. if $p, q \in K$ and $\gamma\colon[0,1] {\rightarrow}M$ is any geodesic with $\gamma(0) = p$ and $\gamma(1) = q$, then $\gamma( [0,1]) \subset K$.

2.  ${M \backslash K}$ satisfies the following curvature assumption: $$\begin{aligned}
    \label{curvasmp}
    -\infty < -b^2 \leq \sec( {M \backslash K}) \leq -a^2 < 0.\end{aligned}$$

Then $K$ is an essential subset of $M$ and ${M^{*}}:= M \cup {M(\infty)}$ is a geometric compactification of $M$ as a topological manifold with boundary. The boundary is homeomorphic to $\partial K$. Further, ${M^{*}}$ is endowed with the structure of a $C^{a/b}$ manifold with boundary, independent of the choice of $K$. \[thm:main\]

Since any point $x$ in a Cartan-Hadamard manifold $M$ may be regarded as a pole, any small closed ball about $x$ is easily seen to be an essential subset for $M$. Therefore Theorem \[thm:main\] generalizes and strengthens the Anderson-Schoen result, for it allows for much greater variety in the topology of $M$; essential subsets relax the stringent 