1
$\begingroup$

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.

Thanks in advance.

  • 2
    The system of equations for a geodesic involves first derivatives of the metric coefficients, so if the metric itself is not real analytic, I wouldn't expect the geodesics to be either.2012-04-06

0 Answers 0