Geodesic
On holomorphic isometries for the Kobayashi and Carathéodory distances on complex manifolds
Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 83 (1989) no. 1, pp. 139-145

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Zbl MR
It is shown that under certain conditions every holomorphic isometry for the Carathéodory or the Kobayashi distances is an isometry for the corrisponding metrics. These results are used to give a characterization of biholomorphic mappings between convex domains and complete circular domains. 
Venturini, Sergio. On holomorphic isometries for the Kobayashi and Carathéodory distances on complex manifolds. Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 83 (1989) no. 1, pp. 139-145. http://geodesic.mathdoc.fr/item/RLINA_1989_8_83_1_a22/
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     title = {On holomorphic isometries for the {Kobayashi} and {Carath\'eodory} distances on complex manifolds},
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