Abstract
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.
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Gaussier, H., Seshadri, H. Totally geodesic discs in strongly convex domains. Math. Z. 274, 185–197 (2013). https://doi.org/10.1007/s00209-012-1063-3
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DOI: https://doi.org/10.1007/s00209-012-1063-3