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The geometry of $\mathbb {C}^2$ equipped with Warren’s metric

Volume 121 / 2018

Szymon Myga Annales Polonici Mathematici 121 (2018), 243-249 MSC: Primary 32Q15. DOI: 10.4064/ap180618-28-8 Published online: 12 October 2018


The aim of this note is to describe the geometry of $\mathbb {C}^2$ equipped with a Kähler metric defined by Warren. It is shown that with that metric $\mathbb {C}^2$ is a flat manifold. Explicit formulae for geodesics and the volume of the geodesic ball are also computed. Finally, a family of similar flat metrics is constructed.


  • Szymon MygaInstitute of Mathematics
    Jagiellonian University
    Łojasiewicza 6
    30-348 Kraków, Poland

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