Condition of elementary flatness and the two-particle Curzon solution

The condition of elementary flatness — used in demonstrating the existence of a strut (along r = 0) in the two-particle Curzon solution of the Einstein field equations — is given a rigorous foundation. We show, by using the Gauss-Bonnet theorem, that if elementary flatness is violated then the spacetime including r = 0 is not a lorentzian manifold.