# Distance function and Hessian in Lorentzian geometries with positive curvature

Suppose $$(M,g)$$ is a Lorentzian manifold with signature $$(-,+,\ldots,+)$$ and a positive curvature. Let $$p \in M$$. Let $$U$$ be a sufficiently small neighborhood of $$p$$ in the exterior of the double null cone emanating from the point $$p$$. Given each $$q \in U$$ we define $$r(q)$$ to be the geodesic distance of the point $$q$$ to $$p$$ in $$U$$.

Is it true that given any $$q \in U$$ and $$X \in T_qM$$ with $$\langle X,X\rangle_g=0$$ and $$\langle X,\nabla r\rangle_g=0$$, there holds $$\mathrm{Hess}\, r(X,X)\geq 0$$?

I am trying to use a Hessian comparison theorem here, but have not been successful since $$X$$ is a null-vector.