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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.

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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. – treble Apr 6 '12 at 18:58

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