Recently, when I read the Proposition 1 of S.Alexander's paper Locally Convex Hypersurfaces of Negatively Curved Spaces. In this proof:
Proposition 1. Let $i: M \rightarrow N$ be a hypersurface immersion of a compact, connected, orientable manifold $M$ of dimension $m \geqslant 2$, and $Z$ be a continuous unit normal. If the sectional curvatures of $N$ satisfy $K_N \leqslant-k<0$ and $Z$ may be chosen so that the eigenvalues of the second fundamental form $S_Z$ satisfy $\lambda \geqslant-\sqrt{k}$, then $M$ is diffeomorphic to the sphere $S^m$.
Proof. Note that here $S_Z$ is defined by $S_Z(X, Y)=\left\langle\nabla_X Z, Y\right\rangle$, for vector fields $X, Y$ on $M$. Since $M$ is compact, $i(M)$ lies in some metric ball $B$. Since $K_N \leqslant 0, \exp _y$ is a diffeomorphism onto $N$, for any $y \in N$, and metric balls are convex. Therefore for any $x \in M$, the geodesic ray $j_x$ in $N$ with initial direction $Z(x)$ strikes $\partial B$ once and transversely, say at the point $p(x)$. If $M$ has no focal points on $j_x$, then the map $p: M \rightarrow \partial B$ is a diffeomorphism, since $\partial B$ is diffeomorphic to $S^m$.
He said that it's sufficient to show that $M$ has no focal points. In my opinion, "focal point" is a local definition, which measures the local behavior of the normal geodesics of the submanifold. By the Rauch comparison theorem acording to Bishop, we can show that $M$ has no focal points. But I think we should prove that globally, the metric ball $B$ can be chosen inside the cut locus of $M$ or $M$ has empty cut locus using local convexity. Excuse me, is this extra? Or am I wrong about the focal points?