Let $M$ be a 2-dimensional Riemannian manifold of non-positive curvature everywhere, of genus > 1. Let $\textbf{D} \subset \textbf{C}$ be the open unit disc in the complex plane, the universal cover of $M$. Let $\gamma \subset \textbf{D}$ be a geodesic lying entirely on a region of zero curvature. It seems to me, because the definition of geodesic is local, that unless $\gamma$ is tangent to some region $A \subset \textbf{D}$ of negative curvature, $\gamma$ will be a Euclidean line through $\textbf{D}$. Is this correct?

Secondly, does anybody have any references that I could peruse to learn how a geodesic $\gamma$ which does in fact pass tangent to some $A$ of negative curvature reacts to this region (will it turn into $A$, away from it, etc. and maybe some way of calculating the actual effect)?

Thank you.

Let $M$ be a 2-dimensional Riemannian manifold of non-positive curvature everywhere, of genus > 1. Let $\textbf{D} \subset \textbf{C}$ be the open unit disc in the complex plane, the universal cover of $M$. Let $\gamma \subset \textbf{D}$ be a curve representing a geodesic in $M$ which is entirely in a region of zero curvature. It seems to me, because the definition of geodesic is local, that unless $\gamma$ is tangent to some region $A \subset \textbf{D}$ of negative curvature, $\gamma$ will be a Euclidean line through $\textbf{D}$. Is this correct?

Secondly, does anybody have any references that I could peruse to learn how a geodesic $\gamma$ which does in fact pass tangent to some $A$ of negative curvature reacts to this region (will it turn into $A$, away from it, etc. and maybe some way of calculating the actual effect)?

Thank you.