A Interpretation of covariant derivative of a vector field

[Pentaquark5]

On Riemannian manifolds $\mathcal{M}$ the covariant derivative can be used for parallel transport by using the Levi-Civita connection. That is

Let $\gamma(s)$ be a smooth curve, and $l_0 \in T_p\mathcal{M}$ the tangent vector at $\gamma(s_0)=p$. Then we can parallel transport $l_0$ along $\gamma$ by using
$$ \nabla_{\gamma'(s)}l(s)=0 .$$
However, I'm having trouble applying this knowledge to the following case (quote from my script):

Let $e_0:=\dot{\gamma}$ and $e_1 \perp e_0$ be unit vectors. Let further $S$ be the $2$-dimensional submanifold of $\mathcal{M}$ obtained by shooting null-geodesics with initial direction $l(0):=e_0+e_1$ from all points along the curve $\gamma$.
Imposing affine parameterisation, this defines on $S$ a vector field $l$ tangent to those geodesics, solution of the equation
$$\nabla_l l=0.$$
I interpret this equation as the parallel transport of $l$ along $l$, but that does not really coincide with the description given in my script.

Could anybody help me make sense of this equation? Thanks

 

[Orodruin]

I interpret this equation as the parallel transport of lll along lll, but that does not really coincide with the description given in my script.

Why not? A geodesic is a curve whose tangent vector is parallel transported along the curve itself.

Also, you seem to be dealing with a pseudo-Riemannian manifold, not a Riemannian manifold. A Riemannian manifold does not have any non-zero null vectors.

 

[Pentaquark5]

Ah, thank you I don't know what went wrong in my head there. And you are of course correct in noting that $\mathcal{M}$ is pseudo Riemannian.

Thanks!