[Apologies if this question is not considered research level, but it received no substantive comments and no answers at math.SE; I thought it was straightforward, but maybe it isn't.]

Part I - What are the formal conditions for the solubility of the geodesic equation

$$\frac{d^2 x^\mu}{ds^2} =- \Gamma^{\mu}_{\alpha \beta} \frac{dx^\alpha}{ds} \frac{dx^\beta}{ds} ?$$

Part II - Consider (see figure) the particular case of a pseudo-Riemannian manifold whose metric is everywhere constant except on the surface $\Sigma$ (and the metric on $\Sigma$ may vary *smoothly* from place to place) such that every causal curve from $p$ to $q$ intersects the surface $\Sigma$ at a single point (except possibly in the null case), which surface - appearances in the figure not withstanding - might be or have null regions, and across which the metric is discontinuous.

Are there *any* conditions on the metric on $\Sigma$ (or otherwise) that make the geodesic equation soluble? (Possibly piecewise?).

**Notes**

The usual approach to solving the geodesic equation will not work, I believe, because of the curvature discontinuity introduced by $\Sigma$ that entails undefined derivatives across $\Sigma$, but maybe if one took as an assumption that $\Sigma$ is nowhere null, one could consider the curves to $\Sigma$ on each side separately and seek identity of their limit points.

If there are regions of $\Sigma$ that are null it seems to me it might be possible for the limit points of the two segments of $\gamma$ to be separated in $\Sigma$.

PS I'm not a mathematician, so in attempting to be suitably precise for those who are, I may in fact have had the opposite effect; I hope others can read between the lines.

parallel(which is the usual replacement for "constant" in the covariant setting), then a metric is always parallel with respect to its own Levi-Civita connection. Also, if your metric is not even continuous, it not at all clear what your differential equation even means. Are you also for a way to even interpret it in your situation? $\endgroup$ – Matthias Ludewig Nov 28 '16 at 19:58