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Geodesics for convex complex ellipsoids. (English) Zbl 0812.32010
Denote by $${\mathcal E} (p)$$ the complex ellipsoid $$\{(z_ 1, \dots, z_ n) \in \mathbb{C}^ n |\;\sum^ n_{j=1} | z_ j |^{2p_ j} < 1\}$$. This paper describes all complex geodesics $$\varphi:E\to{\mathcal E}(p)$$, where $$E:=\{\lambda\in\mathbb{C}|\;|\lambda|<1\}$$. The holomorphic mapping $$\varphi : E \to {\mathcal E} (p)$$ is a complex geodesic if it preserves the Kobayashi distance.
Remark. A simplification of the proof of the main theorem is in the preprint by the first two authors entitled “Geodesics for convex complex ellipsoids. II”.

##### MSC:
 32F45 Invariant metrics and pseudodistances in several complex variables
##### Keywords:
complex ellipsoid; complex geodesics; Kobayashi distance
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##### References:
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