Annali di Matematica Pura ed Applicata

, Volume 154, Issue 1, pp 385–402

Pseudodistances and Pseudometrics on real and complex manifolds

  • Sergio Venturini


In this paper we study the relationships between a class of distances and infinitesimal metrics on real and complex manifolds and their behavior under differentiable and holomorphic mappings. Some application to Riemannian and Finsler geometry are given and also new proofs and generalizations of some results of Royden, Harris and Reiffen on Kobayashi and Carathéodory metrics on complex manifolds are obtained. In particular we prove that on every complex manifold (finite or infinite- dimensional) the Kobayashi distance is the integrated form of the corresponding infinitesimal metric.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    T. J. Barth,The Kobayashi indicatrix at the center of a circular domain, Proc. Amer. Math. Soc.,88 (1983), pp. 527–530.Google Scholar
  2. [2]
    H. Busemann -W. Mayer,On the foundation of the calculus of variations, Trans. Amer. Math. Soc,49 (1941), pp. 173–198.Google Scholar
  3. [3]
    Dugundy,General Topology, Allyn and Bacon, Inc., Boston (1966).Google Scholar
  4. [4]
    H. Federer,Geometric Measure Theory, Springer-Verlag, Berlin (1969).Google Scholar
  5. [5]
    T. Franzoni -E. Vesentini,Holomorphic maps and invariant distances, North Holland, Amsterdam (1980).Google Scholar
  6. [6]
    K. T. Hahn,Some remarks on a new pseudodifferential metric, Ann. Pol. Math.,39 (1981), pp. 71–81.Google Scholar
  7. [7]
    J. Hoffman Jørgensen,The Theory of Analytic Spaces, Various Publication Series, No. 10, Årthus Universitet, Denmark (1970).Google Scholar
  8. [8]
    L. A. Harris,Schwarz-Pick systems of pseudometrics for domains in normed linear spaces, in « Advanced in Holomorphy » (Editor J. A. Barroso), North Holland, Amsterdam (1979), pp. 345–406.Google Scholar
  9. [9]
    J. Kelley,General Topology, Springer-Verlag, New York (1955).Google Scholar
  10. [10]
    S. Kobayashi,Hyperbolic manifolds and holomorphic mappings, Dekker, New York (1970).Google Scholar
  11. [11]
    S.Kobayashi,Transformations Groups in Differential Geometry, Ergebnisse Series, Vol.70, Springer-Verlag (1972).Google Scholar
  12. [12]
    S. Kobatashi,Intrinsic distances, measures and geometric function theory, Bull. of the Amer. Math. Soc.,82 (1976), pp. 357–416.Google Scholar
  13. [13]
    S.Kobayashi,Projective Structures of Hyperbolic Type, Proceedings of 1977 U.S.-Japan Seminar on Minimal Submanifolds, Tokyo (1978).Google Scholar
  14. [14]
    B. O'Byrne,On Finsler geometry and applications to Teichmüller spaces, Ann. Math. Studies,66 (1971), pp. 317–328.Google Scholar
  15. [15]
    H. J. Reiffen,Die Carathéodorysche Distanz und ihre zugehörige Differentialmetrik, Math. Ann.,161 (1965), pp. 315–324.Google Scholar
  16. [16]
    W. Rinow,Die innere geometrie der metrischen raume, Springer-Verlag, Berlin (1961).Google Scholar
  17. [17]
    H. Royden,Remarks on the Kobayashi metrics, inSeveral Complex Variables II, Lect. Notes in Math.,185, Springer-Verlag, Berlin (1971), pp. 125–137.Google Scholar
  18. [18]
    H. Royden,The extension of regular holomorphic maps, Proc. Amer. Math. Soc.,43 (1974), pp. 306–310.Google Scholar
  19. [19]
    H. Wu,Some theorems on projective hyperbolicity, J. Math. Soc. Japan,33 (1981), pp. 79–104.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1989

Authors and Affiliations

  • Sergio Venturini
    • 1
  1. 1.Scuola Normale SuperiorePisaItalia

Personalised recommendations