Suppose we have some Lorentzian manifold $(M,g)$ and at some point $p\in M$ we pick a spacelike 3-dimensional subspace of $T_pM$. We can then expand the vectors in this subspace, using the geodesic equation, to a family of spacelike curves. Locally this gives us a hyper-surface using normal coordinates, but I wonder what happens globally.
- Does the set of points these geodesics pass through form a smooth surface?
- If $(M,g)$ is globally hyperbolic, is this hypersurface (if it exists) Cauchy?
Looking at flat space the above points are obviously true, but I can't think of an argument to lift them to general $M$.
I saw a similar question asked earlier but left unanswered, When are geodesically generated surfaces everywhere spacelike?. I can't really find any theory on 'geodesically generated surfaces' anywhere, any help or further reading material would be greatly appreciated.