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Classification of Holomorphic Functions as Pólya Vector Fields via Differential Geometry

Author

Listed:
  • Lucian-Miti Ionescu

    (Department of Mathematics, Illinois State University, Normal, IL 61790-4520, USA
    These authors contributed equally to this work.)

  • Cristina-Liliana Pripoae

    (Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana 6, RO-010374 Bucharest, Romania
    These authors contributed equally to this work.)

  • Gabriel-Teodor Pripoae

    (Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, RO-010014 Bucharest, Romania
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Lucian-Miti Ionescu & Cristina-Liliana Pripoae & Gabriel-Teodor Pripoae, 2021. "Classification of Holomorphic Functions as Pólya Vector Fields via Differential Geometry," Mathematics, MDPI, vol. 9(16), pages 1-15, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1890-:d:610945
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