Cartan-Hadamard theorem.

The basic idea of an essential subset $K$ is to capture the topology of the manifold $M$ inside $K$ in a such way that the normal geodesic flow off of the boundary of $K$ is a diffeomorphism of the outward normal bundle $Y \times [0, \infty)$ onto $\overline{{M \backslash K}}$. Some sort of hypothesis on the topology and geometry of $M \backslash K$ is necessary in order for the exponential map to be injective, as can be seen by considering the orbit space obtained by taking the upper half space model of the two-dimensional hyperbolic plane under the action of the discrete group of dilations $G = \{ 2^n: n \in {\mathbb{Z}}\}$. Given a sufficiently small ball centred at any point along the ‘waist’ of this quotient space, i.e. the image of $x=0$, one finds normal geodesics off the ball that intersect.

A subset $K \subset M$ is *totally convex in M* if whenever $p, q \in K$ and $\sigma\colon[0,1] \longrightarrow M$ is a geodesic such that $\sigma(0)=p$, $\sigma(1) = q$ then $\sigma([0,1]) \subset K$. The inclusion of $K$ into $M$ is a homotopy equivalence; see [@Klingenberg] for details. It is interesting that totally convex sets play an 