Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Projectively flat Finsler metrics of constant flag curvature

Author: Zhongmin Shen
Journal: Trans. Amer. Math. Soc. 355 (2003), 1713-1728
MSC (2000): Primary 53C60, 53A20
Published electronically: December 2, 2002
MathSciNet review: 1946412
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Finsler metrics on an open subset in ${R}^n$ with straight geodesics are said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). In this paper, we discuss the classification problem on projective Finsler metrics of constant flag curvature. We express them by a Taylor expansion or an algebraic formula. Many examples constructed in this paper can be used as models in Finsler geometry.

References [Enhancements On Off] (What's this?)

  • [AIM] P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, The theory of sprays and Finsler spaces with applications in physics and biology, Fundamental Theories of Physics, vol. 58, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1273129
  • [Al] Ralph Alexander, Planes for which the lines are the shortest paths between points, Illinois J. Math. 22 (1978), no. 2, 177–190. MR 490820
  • [Alv1] Juan Carlos Álvarez Paiva, Contact topology, taut immersions, and Hilbert’s fourth problem, Differential and symplectic topology of knots and curves, Amer. Math. Soc. Transl. Ser. 2, vol. 190, Amer. Math. Soc., Providence, RI, 1999, pp. 1–21. MR 1738387,
  • [Alv2] J. C. Álvarez , Symplectic geometry and Hilbert's fourth problem, preprint.
  • [AlFe] J. C. Álvarez Paiva and E. Fernandes, Crofton formulas in projective Finsler spaces, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 91–100. MR 1655987,
  • [AlGeSm] J. C. Alvarez, I. M. Gelfand, and M. Smirnov, Crofton densities, symplectic geometry and Hilbert’s fourth problem, The Arnold-Gelfand mathematical seminars, Birkhäuser Boston, Boston, MA, 1997, pp. 77–92. MR 1429885
  • [Am] R. V. Ambartzumian, A note on pseudo-metrics on the plane, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37 (1976/77), no. 2, 145–155. MR 0426089,
  • [AZ] H. Akbar-Zadeh, Sur les espaces de Finsler à courbures sectionnelles constantes, Acad. Roy. Belg. Bull. Cl. Sci. (5) 74 (1988), no. 10, 281–322 (French, with English summary). MR 1052466
  • [BaRo] D. Bao and C. Robles, On Randers metrics of constant curvature, Reports on Mathematical Physics (to appear).
  • [BaSh] D. Bao and Z. Shen, Finsler metrics of constant curvature on the Lie group $S^3$, Journal of London Mathematical Society (to appear).
  • [BCS] D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag, New York, 2000. MR 1747675
  • [Be1] L. Berwald, Über eine characteristic Eigenschaft der allgemeinen Räume konstanter Krümmung mit gradlinigen Extremalen, Monatsh. Math. Phys. 36(1929), 315-330.
  • [Be2] L. Berwald, Über die n-dimensionalen Geometrien konstanter Krümmung, in denen die Geraden die kürzesten sind, Math. Z. 30(1929), 449-469.
  • [Bl] W. Blaschke, Integralgeometrie 11: Zur Variationsrechnung, Abh. Math. Sem. Univ. Hamburg, 11(1936), 359-366.
  • [Br1] Robert L. Bryant, Finsler structures on the 2-sphere satisfying 𝐾=1, Finsler geometry (Seattle, WA, 1995) Contemp. Math., vol. 196, Amer. Math. Soc., Providence, RI, 1996, pp. 27–41. MR 1403574,
  • [Br2] Robert L. Bryant, Projectively flat Finsler 2-spheres of constant curvature, Selecta Math. (N.S.) 3 (1997), no. 2, 161–203. MR 1466165,
  • [Br3] R. Bryant, Some remarks on Finsler manifolds with constant flag curvature, Houston J. of Math. 28(2002), 221-262.
  • [Bu] Herbert Busemann, Problem IV: Desarguesian spaces, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R. I., 1976, pp. 131–141. Proc. Sympos. Pure Math., Vol. XXVIII. MR 0430935
  • [Fk1] P. Funk, Über Geometrien, bei denen die Geraden die Kürzesten sind, Math. Annalen 101(1929), 226-237.
  • [Fk2] P. Funk, Über zweidimensionale Finslersche Räume, insbesondere über solche mit geradlinigen Extremalen und positiver konstanter Krümmung, Math. Zeitschr. 40(1936), 86-93.
  • [Fk3] Paul Funk, Eine Kennzeichnung der zweidemensionalen elliptischen Geometrie, Österreich. Akad. Wiss. Math.-Natur. Kl. S.-B. II 172 (1963), 251–269 (German). MR 0171229
  • [Ha] G. Hamel, Uber die Geometrien, in denen die Geraden die Kürtzesten sind, Math. Ann. 57(1903), 231-264.
  • [Hi] D. Hilbert, Mathematical Problems, Bull. Amer. Math. Soc. 37(2001), 407-436. Reprinted from Bull. Amer. Math. Soc. 8 (July 1902), 437-479.
  • [MaSh] M. Matsumoto and H. Shimada, The corrected fundamental theorem on the Randers spaces of constant curvature, Tensor, N.S. (to appear).
  • [Ok] Tsutomu Okada, On models of projectively flat Finsler spaces of constant negative curvature, Tensor (N.S.) 40 (1983), no. 2, 117–124. MR 837784
  • [Po] Aleksei Vasil′evich Pogorelov, Hilbert’s fourth problem, V. H. Winston & Sons, Washington, D.C.; A Halsted Press Book, John Wiley & Sons, New York-Toronto, Ont.-London, 1979. Translated by Richard A. Silverman; Scripta Series in Mathematics. MR 550440
  • [Ran] G. Randers, On an asymmetric metric in the four-space of general relativity, Phys. Rev. 59(1941), 195-199.
  • [Rap] A. Rapcsák, Über die bahntreuen Abbildungen metrischer Räume, Publ. Math. Debrecen 8 (1961), 285–290 (German). MR 0138079
  • [Sh1] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001.
  • [Sh2] Z. Shen, Finsler metrics with $\mathbf{K}=0$ and $\textbf{S}=0$, Canadian J. Math. (to appear).
  • [Sh3] Z. Shen, Two-dimensional Finsler metrics of constant flag curvature, Manuscripta Mathematica (to appear).
  • [Sh4] Z. Shen, Projectively flat Randers metrics of constant flag curvature, Math. Ann. (to appear).
  • [Sz] Z. I. Szabó, Hilbert’s fourth problem. I, Adv. in Math. 59 (1986), no. 3, 185–301. MR 835025,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C60, 53A20

Retrieve articles in all journals with MSC (2000): 53C60, 53A20

Additional Information

Zhongmin Shen
Affiliation: Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216

Received by editor(s): July 1, 2002
Published electronically: December 2, 2002
Article copyright: © Copyright 2002 American Mathematical Society