In this paper, we introduce a class of Finsler metrics with interesting curvature properties. Then we find necessary and sufficient condition under which these Finsler metrics are locally dually flat and Douglas metrics.
1. R. Alexander, Planes for which the lines are the shortest paths between points, Illinois J. of Math., 22(1978), 177-190.
2. R.V. Ambartzumian, A note on pseudo-metrics on the plane, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 37(1976), 145-155.
3. S. B´acs´o and M. Matsumoto, On Finsler spaces of Douglas type, A generalization of notion of Berwald space, Publ. Math. Debrecen. 51(1997), 385-406.
4. L. Berwald, Parallel´ubertragung in allgemeinen R¨aumen. Atti Congr. Intern. Mat Bologna. 4(1928), 263-270.
5. L. Berwald, Uber die n-dimensionalen Geometrien konstanter Kr¨ummung ¨ , in denen die Geraden die k¨urzesten sind, Math. Z. 30(1929), 449-469.
6. W. Blaschke, Integral geometrie 11: Zur Variationsrechnung, Abh. Math. Sem. Univ. Hamburg, 11(1936), 359-366.
7. R. Bryant, Finsler structures on the 2-sphere satisfying K = 1, Finsler Geometry, Contemporary Mathematics 196, Amer. Math. Soc., Providence, RI, 1996, 27-42.
8. P. Funk, Uber Geometrien, bei denen die Geraden die K¨urzesten sind ¨ , Math. Ann. 101(1929), 226-237.
9. P. Funk, Uber zweidimensionale Finslersche R¨aume, insbesondere ¨uber solche mit ger- ¨ adlinigen Extremalen und positiver konstanter Kr¨ummung, Math. Zeitschr. 40(1936), 86-93.
10. P. Funk, Eine Kennzeichnung der zweidimensionalen elliptischen Geometrie, Osterreichische Akad. der Wiss. Math., Sitzungsberichte Abteilung II ¨ 172(1963), 251- 269.
11. E. Guo, H. Liu and X. Mo, On spherically symmetric Finsler metrics with isotropic Berwald curvature. Int. J. Geom. Methods Mod. Phys., 10(10) (2013), 113.
12. G. Hamel, Uber die Geometrieen in denen die Geraden die K¨urzesten sind ¨ , Math. Ann. 57(1903), 231-264.
13. D. Hilbert, Mathematical Problems, Bull. of Amer. Math. Soc. 37(2001), 407-436. Reprinted from Bull. Amer. Math. Soc. 8 (July 1902), 437-479.
14. L Huang and X. Mo, On spherically symmetric Finsler metrics of scalar curvature, J. Geom. Phys., 62(11) (2012), 22792287.
15. L. Huang and X. Mo, Projectively flat Finsler metrics with orthogonal invariance, Ann. Polon. Math. 107(3)(2013), 259270.
16. O. Lungu, Some curvature properties in Randers spaces, Sci. Stud. Res. Ser. Math. Inform. 19(2) (2009), 301308.
17. A.V. Pogorelov, Hilbert’s Fourth Problem, Winston & Wiley, New York, 1982.
18. A. Rapcs´ak, Uber die bahntreuen Abbildungen metrisher R¨aume ¨ , Publ. Math. Debrecen, 8(1961), 285-290.
19. S. F. Rutz, Symmetry in Finsler spaces. Amer. Math. Soc, Providence, RI, 1996. 20. H. Sadeghi, Finsler metrics with bounded Cartan torsion, Journal of Finsler Geometry and its Applications, 2(1) (2021), 51-62.
21. H. Sadeghi, A special class of Finsler metrics, Journal of Finsler Geometry and its Applications, 1(1) (2020), 60-65.
22. H. Sadeghi and M. Bahadori, On projectively related spherically symmetric metrics in Finsler geometry, Journal of Finsler Geometry and its Applications, 1(2) (2020), 39-53.
23. Z. Shen, Projectively flat Finsler metrics of constant flag curvature, Trans of Amer. Math. Soc. 355(4)(2003), 1713-11728.
24. Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, 2001.
25. Z. Shen, Lectures on Finsler Geometry. Singapore, World Scientific, 2001.
26. Z. Shen, Riemann-Finsler geometry with applications to information geometry, Chin. Ann. Math., 27(1) (2006), 73-94.
27. Z. Shen, Projectively related Einstein metrics in Riemann-Finsler geometry, Math. Ann. 320(2001), 625-647.
28. Z. Shen, Funk metrics and R-flat sprays, unpublished (2001).
29. L. Zhou, Projective spherically symmetric Finsler metrics with constant flag curvature in Rn. Geom. Dedicata. 158(1) (2012), 353364.
30. Z.I. Szab´o, Hilbert’s fourth problem, I, Adv. in Math. 59(1986), 185-301.
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