important and somewhat analogous role in the of theory of souls of positively curved manifolds. We again refer the interested reader to [@Klingenberg] and the references therein.

We have the following sufficient condition for an essential subset.

Let $K \subset M$ be a compact Riemannian submanifold with hypersurface boundary $Y$. Suppose that \[thm:characterizationES\]

-   K is totally convex in M, and

-   $\sec({M \backslash K}) \leq 0$.

Then $K$ is an essential subset for $M$.

As $K$ is totally convex, it is also geodesically convex ( i.e. $K$ contains a geodesic segment between any two of its points ). It is well known that $K$ geodesically convex implies that $Y= \partial K$ is convex.

We now check that the image $E(Y\times [0, \infty))$ is a subset of $\overline{{M \backslash K}}$. To see this, notice that any normal geodesic $\gamma_p$ that re-enters $K$ must lie entirely inside $K$ since $K$ is totally convex, but this violates the fact $\gamma_p$ is a geodesic ray with an outward pointing tangent vector at $p$.

Next, Jacobi field estimates and the nonpositive curvature assumption on ${M \backslash K}$ imply that $E$ is a local diffeomorphism on $Y \times [0, \infty)$. We need only argue that 