Barbilian's metrization procedure in the plane yields either Riemannian or Lagrange generalized metrics

Wladimir G. Boskoff; Bogdan D. Suceavă

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 4, page 1059-1068
  • ISSN: 0011-4642

Abstract

top
In the present paper we answer two questions raised by Barbilian in 1960. First, we study how far can the hypothesis of Barbilian's metrization procedure can be relaxed. Then, we prove that Barbilian's metrization procedure in the plane generates either Riemannian metrics or Lagrance generalized metrics not reducible to Finslerian or Langrangian metrics.

How to cite

top

Boskoff, Wladimir G., and Suceavă, Bogdan D.. "Barbilian's metrization procedure in the plane yields either Riemannian or Lagrange generalized metrics." Czechoslovak Mathematical Journal 58.4 (2008): 1059-1068. <http://eudml.org/doc/37885>.

@article{Boskoff2008,
abstract = {In the present paper we answer two questions raised by Barbilian in 1960. First, we study how far can the hypothesis of Barbilian's metrization procedure can be relaxed. Then, we prove that Barbilian's metrization procedure in the plane generates either Riemannian metrics or Lagrance generalized metrics not reducible to Finslerian or Langrangian metrics.},
author = {Boskoff, Wladimir G., Suceavă, Bogdan D.},
journal = {Czechoslovak Mathematical Journal},
keywords = {\{Riemannian metrics; Finslerian metrics; Lagrangian metrics; Lagrange generalized metrics; Barbilian's metrization procedure; Apollonian metric\}; Riemannian metrics; Finslerian metrics; Lagrangian metrics; Lagrange generalized metrics; Barbilian's metrization procedure; Apollonian metric},
language = {eng},
number = {4},
pages = {1059-1068},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Barbilian's metrization procedure in the plane yields either Riemannian or Lagrange generalized metrics},
url = {http://eudml.org/doc/37885},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Boskoff, Wladimir G.
AU - Suceavă, Bogdan D.
TI - Barbilian's metrization procedure in the plane yields either Riemannian or Lagrange generalized metrics
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 1059
EP - 1068
AB - In the present paper we answer two questions raised by Barbilian in 1960. First, we study how far can the hypothesis of Barbilian's metrization procedure can be relaxed. Then, we prove that Barbilian's metrization procedure in the plane generates either Riemannian metrics or Lagrance generalized metrics not reducible to Finslerian or Langrangian metrics.
LA - eng
KW - {Riemannian metrics; Finslerian metrics; Lagrangian metrics; Lagrange generalized metrics; Barbilian's metrization procedure; Apollonian metric}; Riemannian metrics; Finslerian metrics; Lagrangian metrics; Lagrange generalized metrics; Barbilian's metrization procedure; Apollonian metric
UR - http://eudml.org/doc/37885
ER -

References

top
  1. Anastasiei, M., 10.1090/conm/196/02443, Contemp. Math. 196 (1996), 161-169. (1996) Zbl0857.53016MR1403588DOI10.1090/conm/196/02443
  2. Barbilian, D., Einordnung von Lobayschewskys Massenbestimmung in einer gewissen allgemeinen Metrik der Jordansche Bereiche, Časopis Mathematiky a Fysiky 64 (1934-35), 182-183 1.0601.02. (1934) 
  3. Barbilian, D., Asupra unui principiu de metrizare, Stud. Cercet. Mat. 10 (1959), 68-116. (1959) 
  4. Barbilian, D., Fundamentele metricilor abstracte ale lui Poincaré şi Carathéodory ca aplicaţie a unui principiu general de metrizare, Stud. Cercet. Mat. 10 (1959), 273-306. (1959) MR0131208
  5. Barbilian, D., J-metricile naturale finsleriene, Stud. Cercet. Mat. 11 (1960), 7-44. (1960) MR0124869
  6. Barbilian, D., Radu, N., J-metricile naturale finsleriene şi funcţia de reprezentare a lui Riemann, Stud. Cercet. Mat. 12 (1962), 21-36. (1962) MR0144292
  7. Beardon, A. F., The Apollonian metric of a domain in , in Quasiconformal mappings and analysis, Springer-Verlag (1998), 91-108. (1998) MR1488447
  8. Boskoff, W. G., -Riemannian manifolds and Barbilian spaces, Stud. Cercet. Mat. 46 (1994), 317-325. (1994) Zbl0817.53012MR1681684
  9. Boskoff, W. G., Finslerian and induced Riemannian structures for natural Barbilian spaces, Stud. Cercet. Mat. 47 (1995), 9-16. (1995) Zbl0835.53082MR1682536
  10. Boskoff, W. G., The connection between Barbilian and Hadamard spaces, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sr. 39 (1996), 105-111. (1996) Zbl0888.53024
  11. Boskoff, W. G., Hyperbolic geometry and Barbilian spaces, Istituto per la Ricerca di Base, Hardronic Press (1996). (1996) MR1386390
  12. Boskoff, W. G., A generalized Lagrange space induced by the Barbilian distance, Stud. Cercet. Mat. 50 (1998), 125-129. (1998) Zbl1017.53024MR1839634
  13. Boskoff, W. G., Horja, P., The characterization of some spectral Barbilian spaces using the Tzitzeica construction, Stud. Cercet. Mat. 46 (1994), 503-514. (1994) Zbl0822.51003MR1680832
  14. Boskoff, W. G., Suceavă, B. D., 10.1016/j.hm.2006.06.001, Historia Mathematica 34 (2007), 221-224. (2007) MR2320101DOI10.1016/j.hm.2006.06.001
  15. Boskoff, W. G., Suceavă, B. D., The history of Barbilian metrization procedure and Barbilian spaces, Mem. Secţ. Ştiinţ. Acad. Română Ser. IV 28 (2005), 7-16. (2005) MR2360445
  16. Boskoff, W. G., Ciucă, M. G., Suceavă, B. D., Distances induced by Barbilian's metrization procedure, Houston J. Math. 33 (2007), 709-717. (2007) Zbl1140.53011MR2335731
  17. Chern, S. S., Chen, W. H., Lam, K. S., Lectures on Differential Geometry, World Scientific (1999). (1999) Zbl0940.53001MR1735502
  18. Gehring, F. W., Hag, K., 10.1090/conm/256/04003, Contemp. Math. 256 (2000), 143-163. (2000) Zbl0964.30024MR1759676DOI10.1090/conm/256/04003
  19. Hästö, P. A., The Appollonian metric: uniformity and quasiconvexity, Ann. Acad. Sci. Fennicae 28 (2003), 385-414. (2003) MR1996444
  20. Hästö, P. A., 10.1155/S1085337503309042, Abstr. Appl. Anal. (2003), 1141-1158. (2003) MR2041216DOI10.1155/S1085337503309042
  21. Hästö, P. A., 10.1007/BF03321068, Comput. Methods Funct. Theory 4 (2004), 249-273. (2004) Zbl1073.30036MR2147384DOI10.1007/BF03321068
  22. Hästö, P. A., 10.4310/CAG.2004.v12.n4.a7, Comm. Anal. Geom. 12 (2004), 927-947. (2004) MR2104081DOI10.4310/CAG.2004.v12.n4.a7
  23. Hästö, P. A., Lindén, H., 10.1080/02781070410001712702, Complex Var. Theory Appl. 49 (2004), 405-415. (2004) MR2073171DOI10.1080/02781070410001712702
  24. Ibragimov, Z., On the Apollonian metric of domains in , Complex Var. Theory Appl. 48 (2003), 837-855. (2003) MR2014392
  25. Ibragimov, Z., 10.1007/BF03321045, Comput. Methods Funct. Theory 3 (2003), 397-411. (2003) Zbl1055.30039MR2082025DOI10.1007/BF03321045
  26. Kelly, P. J., 10.2307/2307467, Amer. Math. Monthly 61 (1954), 311-319. (1954) Zbl0055.41002MR0061397DOI10.2307/2307467
  27. Miron, R., Anastasiei, M., Bucătaru, I., The Geometry of Lagrange Spaces Handbook of Finsler geometry, Kluwer Acad. Publ., Dordrecht (2003), 969-1122. (2003) MR2066452

NotesEmbed ?

top

You must be logged in to post comments.