I encountered online the following exercise:

Let $M$ be a Lorentz manifold of $\dim(M)=n$, and let $\psi:\Sigma\to M$ be a spacelike submanifold of dimension $n-2$ embedded into $M$. We will identify $\psi(\Sigma)\subset M$ with $\Sigma$. Let $\gamma:[a,b]\to M$ be a null geodesic with $\gamma(a)=p\in\Sigma$, $\gamma(b)=q\in M\setminus\Sigma$, and $\gamma'(a)\perp T_p\Sigma$. Such a geodesic will be called a null geodesic normal to $\Sigma$. Then we define the null second fundamental form $\chi$ of $\Sigma$ at $p$ relative to $L=\gamma'(a)$ by $$ \chi(X,Y) = \langle \nabla_XY,L\rangle , $$ for $X,Y\in\mathfrak{X}(\Sigma)$. Finally, a point $r=\gamma(s)$ is called conjugate to $\Sigma$ along $\gamma$ if there exists a nontrivial Jacobi field $X\in\mathfrak{X}^\bot(\gamma)$ along $\gamma$ such that $$(i) \ \ \ \ \ \ X(s)=0,$$ $$(ii) \ \ \ \ \ \ X(a)\in T_p\Sigma,$$ and $$(iii) \ \ \ \ \ \ \langle\nabla_LX,V\rangle=-\chi(X,V), \ \forall V\in T_p\Sigma.$$ In the exercise, one is asked to prove that if there exists a point $\gamma(s)$ conjugate to $\Sigma$ along $\gamma$ for some $s\in(a,b)$, then there exists a strictly timelike curve connecting $\Sigma$ and $q$.

I would appreciate any tips in order to start the problem properly. Intuitively, this makes sense when I'm making a picture of the set up because we can clearly deform the null geodesic in a timelike one, but I have a lot of difficulties to formalize my idea and I'm a bit struggling to understand why we are defining the null second fundamental form and how I should use it.

  • $\begingroup$ The intuitive idea is correct. You can see the detail in Penrose "Techniques of differential topology in relativity.". See theorem 7.27 $\endgroup$ – MBN Apr 12 at 6:02

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