# Geodesically generated surfaces

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.

• I think you need completely geodesic or something like that, but that is of course a global condition not available at point $p$. Otherwise, there may be nontrivial self-intersection, for example. – user10354138 Oct 11 '18 at 14:06
• Obviously self-intersection is a possibility, but I don't think it rules out te possibility of the spanned space being a surface. – Berend Visser Oct 12 '18 at 10:10