Following situation:
Let $(M,g)$ be a globally hyperbolic spacetime, $S \subset M$ Cauchy hypersurface with unit normal vector field $n$, $\Sigma \subset S$ compact 2-dim. submanifold with unit normal vector field $\nu$ in $S$.Let be the $c_p$ null geodesic with $c^{'}_p(0)=n_p+\nu_p$. Now I have two questions on the definitions: 1)Is the following consideration correct:
$n_p, \nu_p, n_p+\nu_p$ are tangent vectors in $T_pM$ and also $n_p \perp T_pS$, $\nu_p \perp T_p\Sigma$ right?
Now if I look at the picture
and think of the intuitive idea of a normal vector field, then it can't be a tangent and a normal vector at the same time (s. picture)
Do I see it right, that this "contradiction" comes from the fact that here we do not have a Riemannian metric and therefore $n_p$ for example can be a tangent vector and a normal vector at the same time?
2)I'm not sure if I frankly understand what it means that $\nu$ is the normal vector field of $\Sigma$ in $S$. It was explained to me with a picture:
that it means that $\nu$ is "contained" in $S$ but $\nu$ is a vector field, it does not have values in $M$, how can it be then contained in $S$?