Abstract
We review Pólya 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 Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Pólya 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.
Highlights
Student’s learning of Complex Analysis benefits greatly from physical interpretations, as advocated from the “begining” by Felix Klein [1] when writing about the corresponding work done by Riemann
With the integration tool in it, Pólya associated to a complex function f (z) = u + iv a vector field Vf = , which can be interpreted as the velocity field of a fluid flow, and consider its integrability, in order to find such complex potentials
We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the canonical geometry of the plane, are the special Möbius transformations of the form f (z) = z+b d
Summary
Student’s learning of Complex Analysis benefits greatly from physical interpretations, as advocated from the “begining” by Felix Klein [1] when writing about the corresponding work done by Riemann. Beyond their intrinsic role in the geometries of the plane, the Möbius transformations are quite appropriate as test functions in the big family of holomorphic functions [4,5]. 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 This is part of a more general theory, started in [7,8,9], which geometrizates the vector fields through families of associated affine connections or semiRiemannian metrics.
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