Let $(\mathcal{M},g)$ be a geodesically complete Riemannian manifold. So $\forall x \in \mathcal{M}$, the exponential map $\exp_x$ is defined on all of $T_x \mathcal{M}$. I have two related questions:
(1) Is $\exp_x : T_x \mathcal{M} \rightarrow \mathcal{M}$ surjective for all $x \in \mathcal{M}$? If not, under what conditions on $\mathcal{M}$ is $\exp_x$ surjective (e.g., assuming $\mathcal{M}$ is compact)?
(2) If $\exp_x : T_x \mathcal{M} \rightarrow \mathcal{M}$ is surjective, is there a subset of $S_x \subset T_x \mathcal{M}$ such that $\exp_x$ is invertible on $S_x$ and the image of $S_x$ is almost all of $\mathcal{M}$? I.e., $\exp_x(S_x) = \mathcal{M} \setminus U_x$ for some small (e.g., 'measure zero') set $U_x$. For example, consider the unit 2-sphere: $\mathcal{M} = S^2 \subset \mathbb{R}^3$. Then $exp_x : B_{\pi} \rightarrow S^2 \setminus \{-x\}$ is invertible.
1 is is a corrolary of the Hopf Rinow theorem:
See the third condition in the statement of the theorem.
https://en.wikipedia.org/wiki/Hopf%E2%80%93Rinow_theorem