Geodesic completeness for some meromorphic metrics (Q2777999)

The author is concerned with generalizing the ideas of ``metric'' and geodesic for a complex manifold \(M\), following two directions: metrics are somewhere allowed not to be of maximum rank, or to have ``poles'' somewhere else. This metric is, informally speaking, a symmetric quadratic form on the holomorphic tangent space at each point \(p\in M\), holomorphically depending on the point itself. Of course, it couldn't have any ``signature'', but, by symmetry, it induces a canonical Levi-Civita's connexion on \(M\), which in turn allows one to define geodesies to be auto-parallel paths. The author emphasizes that these curves are complex ones. NEWLINENEWLINENEWLINEThe concept of coercivity of a warped product is introduced: informally speaking, it amounts to the fact that primitives of ``square roots'' of some rational functions of the coefficients involved in the metric can be analytically continued until they take all complex values but at most a finite number of ones. Geodesies show various types of ``singularities'': the author records, among the other ones, ``logarithmic'' singularities: they are, more or less, points resembling 0 in connection with \(z\mapsto\log z\); rather more formally, a ``logarithmic singularity'' \(\ell\) is a point in a two dimensional real topological manifold, admitting a neighbourhood \(\mathcal U\) such that \(\mathcal U\setminus\{\ell\}\) is a Riemann surface, but there is no complex structure ``at'' \(\ell\). This type of singularity arises from the fact that geodesic equations admit first integrals whose solutions have poles with nonzero residues. NEWLINENEWLINENEWLINEThe notion of completeness is introduced: a path is essentially a holomorphic function \(F : S\to M\), where \(S\) is a Riemann surface over a region of \(\mathbb P^1\), admitting a projection mapping \(\pi: S\to\mathbb P^1\): it is complete provided that \(\mathbb P^1 \setminus\pi(S)\) is a finite set. Completeness theorems are given in the framework of warped products of Riemann surfaces. NEWLINENEWLINENEWLINEThe main result of the paper is the following theorem: A warped product of Riemann surfaces is complete (i.e. ``almost every '' geodesic is complete) iff it is coercive. A wide class of coercive direct manifolds is given.