# Hopf-Rinow theorem

If $(M,g)$ is a riemannian manifold. $M$ is complete(geodesically) then any two point can be joined by a geodesic.

Geodesic($\gamma(t))$ is smooth curve such that $\nabla_{\gamma'^{(t)}}\gamma'(t)=0$. Where $\nabla$ is levi civita connection for metric $g$.

Now question is: Whether always these curve(geodesic) is real analytic??? Hopf-Rinow theorem proves that there is a smooth curve which is goedesic which joins two points.