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I have a very basic question on the definition of foliations and geodesic congruences.

My understanding is that:

  • Geodesic congruences are families of geodesics such that locally, every point belongs to exactly one geodesic.
  • Foliations an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension $p$.

One important distinction seems to be that foliations are defined on the entire manifold whereas geodesic congruences can be on any open subregion.

For instance, a trivial example when the two coincide is when we have a family of parellel lines in $\mathbb{R}^2$. We can also have the collection of axial circles on the torus. Unless I am mistaken, the collection of curves in both of these examples satisfy the definitions of geodesic congruences and foliations.

My question: if given a foliation of a manifold by geodesics, do we have a geodesic congruence? The answer seems to be in the affirmative, and (at least in terms of the picture I have in my head) the two notions seem very closely related, however I have not found any resources that talk about both of them.

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I think so. Geodesics are always connected immersed submanifolds of dimension $1$. The fact that they form a foliation means that each geodesic is an equivalence class of the relation, so that in particular every point of the manifold belongs to a unique geodesics. The definition of (here global) congruence of geodesics is satisfied. Actually, one should also check the smoothness requirements involved in the definitions but I think they are fulfilled.

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