Suppose one has a geodesically complete pseudo-Riemannian manifold $M$ i.e. the exponential map is defined for all tangent vectors on the manifold. Can one make a sensible statement about whether (or when) the exponential map restricted to some arbitrary tangent space

\begin{equation} \exp : \text{T}_p M \to M \end{equation}

for some $p \in M$ will be surjective?