# Geodesics of null surface and induced connection

Although this is a question related to GR this is purely a math question, so I think this is the right place to ask it.

Let $$(M,g)$$ be a Lorentzian manifold. Let $$\phi : \Sigma\to M$$ be one null hypersurface, by which we mean its normal is null.

It is common in GR to argue the following:

1. Since the normal vector $$\zeta$$ to $$\Sigma$$ is null, it is orthogonal to itself. Hence the normal to $$\Sigma$$ also lives in its tangent bundle.

2. In that case, one can consider its integral lines. One works formally to show that $$\zeta$$ satisfies the autoparallel equation $$\zeta^\mu \nabla_\mu\zeta_\nu=\alpha\zeta_\nu.$$

Now, I have one problem here. It seems intuitively clear to me that being a geodesic on a submanifold is not the same as being a geodesic on the ambient space.

One example that comes to my mind is in $$\mathbb{R}^3$$. The geodesics are straight lines. If one looks to the submanifold $$S^2$$, the geodesics are great circles which as seen on curves on $$\mathbb{R}^3$$ are not geodesics.

The central question seems to be: what is the connection with respect to which geodesics on $$\Sigma$$ are being considered, and how it connects to the Levi-Civita connection on $$M$$?

Is it the pullback connection? Or is it some sort of "projected" connection, as when we "project the covariant derivative to the tangent space of $$S^2$$" in $$\mathbb{R}^3$$? How it relates to the induced metric $$\phi^\ast g$$?

In short: when physicists consider these generators of the null surface, which are integral lines of the normal as seen as a tangent vector, what is the appropriate way to make sense of the resulting geodesics? How to make this whole discussion rigorous?