A holomorphic point of view about geodesic completeness

We propose to apply the idea of analytical continuation in the complex domain to the problem of geodesic completeness. We shall analyse rather in detail the cases of analytical warped products of real lines, these ones in parallel with their complex counterparts, and of Clifton-Pohl torus, to show that our definition sheds a bit of new light on the behaviour of 'singularities' of geodesics in space-time. We also show that some geodesics, which 'end' at finite time in the classical sense, can be naturally continued besides their ends. As a matter of fact, complex metrics naturally show a meromorphic behaviour, or a degenerating one, so we shall study also this fact in detail.