If a manifold $M$ admits a global chart, does this imply that there exists a point $p\in M$ such that $Cut_p=\emptyset$?
Recall: Definition of $Cut_p$:
Let $\mathfrak{C}_p$ be defined as the set of all vectors v in $T_pM$ such that $\gamma(t)=\exp_p(tv)$ is a minimizing geodesic for each $t \in [0,1] $ but fails to be a unique minimizing for $t \in [0,1 + \epsilon)$ for each $\epsilon > 0$ (Note: uniqueness is important because on $S^1$ the cut-locus is the antipodal point).
We define $Cut_p:=Exp_p(\mathfrak{C}_p)$ and call this the cut locus of $p$ in $M$.