Suppose $X$ is a Riemann surface and $ a\in\ X $ suppose $ \phi\in\mathcal O_a $ is a holomorphic function germ at $a.$ According to the theorem 7.8 of Forster's book Lectures on Riemann surfaces on page 47 $ \phi $ has a maximal analytic continuation $(Y,p,f,b).$
Now suppose $ \phi\in\mathcal O_a $ admits analytical continuation along every curve in $X$ which starts at $a$.Is it always true that $ p:Y\rightarrow X $ has path lifting property.
If I assume another extra condition that , $X$ is simply connected then Monodormy theorem imply that $p:Y\rightarrow X $ has path lifting property.
Another question is that whether $p:Y\rightarrow X $ is covering or not?
Obviously path lifting property of $p:Y\rightarrow X$ implies that $p:Y\rightarrow X$ is a covering since $p:Y\rightarrow X$ is unbranched holomorphic according to the definition. Conversely any covering map has path lifting property.
Definition of path lifting property of a map given in the book of Otto Forster's book on page 25 is that - A continuous map $p:Y\rightarrow X$ is said to have path lifting property if for every path $u:[0,1]\rightarrow X$ and for every point $y_0 \in Y$ with $p(y_0)=u(0)$ there exists a lifting $\tilde u :[0,1]\rightarrow Y$ of $u$ such that $\tilde u(0)=y_0$.