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Transactions of the American Mathematical Society

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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
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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.

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  • [AIM] P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, The theory of sprays and Finsler spaces with applications in physics and biology, FTPH 58, Kluwer Academic Publishers, 1993. MR 95e:53094
  • [Al] R. Alexander, Planes for which the lines are the shortest paths between points, Illinois J. of Math., 22(1978), 177-190. MR 82d:53042
  • [Alv1] J. C. Álvarez Paiva, Contact topology, taut immersions, and Hilbert's fourth problem, in ``Differential and Symplectic Topology of Knots and Curves''. S. Tabachnikov (Ed.). Adv. in Math. Sciences, Amer. Math. Soc., Providence, RI, 1999, 1-21. MR 2001f:53161
  • [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, Electronic Research Announcements of the Amer. Math. Soc. 4(1998), 91-100. MR 99j:53097
  • [AlGeSm] J. C. Álvarez, I. M. Gelfand and M. Smirnov, Crofton densities, symplectic geometry and Hilbert's fourth problem, Arnold-Gelfand Mathematical Seminars, Geometry and Singularity Theory, V .I. Arnold, I. M. Gelfand, M. Smirnov, and V. S. Retakh (eds.). Birkhäuser, Boston, 1997, pp. 77-92. MR 98a:52005
  • [Am] R. V. Ambartzumian, A note on pseudo-metrics on the plane, Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 37(1976), 145-155. MR 54:14035
  • [AZ] H. Akbar-Zadeh, Sur les espaces de Finsler à courbures sectionnelles constantes, Bull. Acad. Roy. Bel. Cl, Sci, 5e Série - Tome LXXXIV (1988), 281-322. MR 91f:53069
  • [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, Springer, 2000. MR 2001g:53130
  • [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] R. Bryant, Finsler structures on the 2-sphere satisfying $K=1$, Finsler Geometry, Contemporary Mathematics 196, Amer. Math. Soc., Providence, RI, 1996, 27-42. MR 97e:53128
  • [Br2] R. Bryant, Projectively flat Finsler $2$-spheres of constant curvature, Selecta Math., New Series, 3(1997), 161-204. MR 98i:53101
  • [Br3] R. Bryant, Some remarks on Finsler manifolds with constant flag curvature, Houston J. of Math. 28(2002), 221-262.
  • [Bu] H. Busemann, Problem IV: Desarguesian spaces, in Mathematical Developments arising from Hilbert Problems, Proc. Sympos. Pure Math. 28(1976), Amer. Math. Soc., Providence, RI, 131-141. MR 55:3940
  • [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] P. Funk, Eine Kennzeichnung der zweidimensionalen elliptischen Geometrie, Öster- reichische Akad. der Wiss. Math.-Natur., Sitzungsberichte Abteilung II 172(1963), 251-269. MR 30:1460
  • [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] T. Okada, On models of projectively flat Finsler spaces of constant negative curvature, Tensor, N. S. 40(1983), 117-123. MR 87c:53124
  • [Po] A. V. Pogorelov, Hilbert's Fourth Problem, Winston & Wiley, New York, 1979. MR 80j:53066
  • [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. MR 25:1526
  • [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), 185-301. MR 88f:53113

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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

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