Classification of Holomorphic Functions as Pólya Vector Fields via Differential Geometry

We review Polya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Polya vector field is torse-forming in the cannonical geometry of the plane, are the special Mobius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Polya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields.