# Existence of a strictly a timelike curve on Lorentzian manifold

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.

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