self intersection of two Lorentzian manifolds also Lorentzian in this case?

Consider a manifold $zeta:={varphi_S}cap{varphi_T}$ as the non-empty transversal intersection between two Lorentzian manifolds, s.t. all three manifolds have dimension $D=3+1.$ Is $zeta$ also a Lorentzian manifold? I know that $zeta$ must be a manifold due to the transversality condition, but I am not sure whether it is Lorentzian.