Geodesic completeness for some meromorphic metrics

In this paper we shall be concerned with generalizing the ideas of ’metric’ and geodesic for a complex manifold M : we emphasize that our curves will be complex ones; a metric will be, informally speaking, a symmetric quadratic form on the holomorphic tangent space at each point p ∈ M , holomorphically depending on the point itself; of course, it couldn’t have any ’signature’, but, by simmetry, it induces a canonical Levi-Civita’s connexion on M , which in turn allows us to define geodesics to be auto-parallel paths. We illustrate some motivations (see [DNF] p.186 ff): consider the space F of antisymmetric covariant tensors of rank two in Minkowski’s space R1,3: electromagnetic fields are such ones. Let F ∈ F : we can write F = i<j Fijdx ∧ dx where x...x are the natural coordinate functions on R1,3. At each point, the space Fp of all tensors in F evaluated at p is a sixdimensional real vector space; moreover, the adjoint operator ∗ with respect to Minkowski’s metric is such that ∗∗ = −1: all these facts imply that Fp could be thought of as a complex three dimensional vector space Gp by setting (a+ib)F = aF+b∗F . Now ∗ is SO(1, 3)−invariant, hence SO(1, 3) is a group of (complex) linear transformations of Gp, preserving the quadratic form 〈F, F 〉 = − ∗ (F ∧ (∗F ) + iF ∧ F ): this means that this ’norm’ is invariant by Lorentz transformations, hence it is of relevant physical interest. If we introduce the following coordinate functions on Gp: z = F01 − iF23, z 2 = F02 + iF13 and z 1 = F03 − iF12, we have that 〈F, F 〉 = (z) + (z) + (z), hence there naturally arises the so called complex-Euclidean metric on C: on one hand, by changing coordinates we are brought to a generic symmetric bilinear form on C; on the other one there arise ’poles’ if