On holomorphic isometries for the Kobayashi and Carathéodory distances on complex manifolds

Sergio Venturini

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti (1989)

  • Volume: 83, Issue: 1, page 139-145
  • ISSN: 0392-7881

Abstract

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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.

How to cite

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Venturini, Sergio. "On holomorphic isometries for the Kobayashi and Carathéodory distances on complex manifolds." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 83.1 (1989): 139-145. <http://eudml.org/doc/289318>.

@article{Venturini1989,
abstract = {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.},
author = {Venturini, Sergio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti},
keywords = {Complex manifolds; Convex and complete circular domains; Carathéodory and Kobayashi distances and metrics},
language = {eng},
month = {12},
number = {1},
pages = {139-145},
publisher = {Accademia Nazionale dei Lincei},
title = {On holomorphic isometries for the Kobayashi and Carathéodory distances on complex manifolds},
url = {http://eudml.org/doc/289318},
volume = {83},
year = {1989},
}

TY - JOUR
AU - Venturini, Sergio
TI - On holomorphic isometries for the Kobayashi and Carathéodory distances on complex manifolds
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
DA - 1989/12//
PB - Accademia Nazionale dei Lincei
VL - 83
IS - 1
SP - 139
EP - 145
AB - 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.
LA - eng
KW - Complex manifolds; Convex and complete circular domains; Carathéodory and Kobayashi distances and metrics
UR - http://eudml.org/doc/289318
ER -

References

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  1. BARTH, T.J., 1983. The Kobayashi indicatrix at the center of a circular domain. Proc. Amer. Math. Soc, 88: 527-530. Zbl0494.32008MR699426DOI10.2307/2045007
  2. DINEEN, S. - TIMONEY, R.M. - VIGUÉ, J.P., 1985. Pseudodistances invariantes sur les domaines d'un espace localement convexe. Annali Scuola Normale Superiore Pisa, (4), 12: 515-529. Zbl0603.46052MR848840
  3. FRANZONI, T. - VESENTINI, E., 1980. Holomorphic maps and invariant distance. North Holland, Amsterdam. Zbl0447.46040MR563329
  4. HARRIS, L.A., 1979. Schwarz-Pick systems of pseudometrics for domains in normed linear spaces. In «Advances in Holomorphy» (Editor J.A. BARROSO), North Holland, Amsterdam: 345-406. MR520667
  5. KOBAYASHI, S., 1970. Hyperbolic manifolds and holomorphic mappings. Dekker, New York. Zbl0207.37902MR277770
  6. KOBAYASHI, S., 1976. Intrinsic distances, measures and geometric function theory. Bull, of the Amer. Math. Soc., 82: 357-416. Zbl0346.32031MR414940DOI10.1090/S0002-9904-1976-14018-9
  7. LEMPERT, L., 1981. La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. France, 109: 427-474. Zbl0492.32025MR660145
  8. LEMPERT, L., 1982. Holomorphic retracts and intrinsic metrics in convex domains. Analysis Mathematica, 8: 257-261. Zbl0509.32015MR690838DOI10.1007/BF02201775
  9. PATRIZIO, G., 1986. On holomorphic maps between domains in C n . Ann. Scuola Normale Superiore, (4), 12: 267-279 Zbl0611.32022MR876125
  10. REIFFEN, H. J., 1963. Die differentialgeometrischen eigenschaften der invarianten distanzfunktion von Carathéodory. Schr. Math. Inst. Univ. Munster, 26, Munster. Zbl0115.16303MR158093
  11. RINOW, W., 1961. Die innere geometrie der metrischen raume. Springer-Verlag, Berlin. Zbl0096.16302MR123969
  12. ROYDEN, H., 1971. Remarks on the Kobayashi metrics. In «Several Complex variables II», Lect. Notes in Math.185, Springer-Verlag, Berlin: 125-137. MR304694
  13. ROYDEN, H. - WONG, P.M., Carathéodory and Kobayashi metric on Convex Domains. Preprint. 
  14. VESENTINI, E., 1982. Complex geodesies. Compositio Math., 44: 375-394. MR662466
  15. VIGUÉ, J.P., 1982. Sur les applications holomorphes isométriques pour la distance de Carathéodory. Annali Scuola Normale Superiore, (4), 9: 255-261. Zbl0507.32017MR674974
  16. VIGUÉ, J.P., 1984. Caractérisation des automorphismes analytiques d'un domain convexe borné. C. R. Acad. Sc. Paris, 299: 101-104. Zbl0589.32042MR756530
  17. VIGUÉ, J.P., Sur la caractérisation des automorphismes analitiques d'un domain borné. To appear. MR911450

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