Holomorphic ( maybe degenerate ) geodesic completeness

In this paper we investigate possible extensions of the idea of geodesic completeness in complex manifolds, following two directions: ’metrics’ are meant to be complex-holomorphic (or meromorphic) symmetric sections of the doubly covariant holomorphic tensor bundle, somewhere allowed not to be of maximum rank, and geodesics are defined on Riemann surfaces which are domains over regions in the complex plane. In this setting the notion of meromorphic metric connection is introduced. Completeness theorems are given about two and three dimensional complex manifolds. 1 Analytic continuation In this section we briefly generalize the notion of analytic continuation regarding holomorphic functions with values in general complex manifolds: this setting is completely analogous to classical complex analysis, thus proofs will be omitted; the reader could refer to [CAS] or [SPR]. We generalize also the notion of path, to be defined on one dimensional complex manifolds, instead of real ones, that is to say open intervals. Definition 1 Let M be a connected complex manifold: a path element in M is a pair (U, f), where U is a connected open set in C and f a holomorphic function defined on U with values in M . Two function elements (U, f) and (V, g) in M may be connected if there exists a finite sequence