Mathematische Zeitschrift

, Volume 274, Issue 1–2, pp 185–197 | Cite as

Totally geodesic discs in strongly convex domains

  • Hervé GaussierEmail author
  • Harish Seshadri


We prove that every isometry from the unit disk Δ in \({\mathbb{C}}\) , endowed with the Poincaré distance, to a strongly convex bounded domain Ω of class \({\mathcal{C}^3}\) in \({\mathbb{C}^n}\) , endowed with the Kobayashi distance, is the composition of a complex geodesic of Ω with either a conformal or an anti-conformal automorphism of Δ. As a corollary we obtain that every isometry for the Kobayashi distance, from a strongly convex bounded domain of class \({\mathcal{C}^3}\) in \({\mathbb{C}^n}\) to a strongly convex bounded domain of class \({\mathcal{C}^3}\) in \({\mathbb{C}^m}\) , is either holomorphic or anti-holomorphic.


Complex Manifold Convex Domain Pseudoconvex Domain Distributional Derivative Complex Geodesic 
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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.UJF-Grenoble 1, Institut Fourier, CNRS UMR 5582GrenobleFrance
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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