We provide an extension of Obata's theorem to Finsler geometry and establish some rigidity results based on a second-order differential equation. Mainly, we prove that every complete simply connected Finsler manifold of positive constant flag curvature is isometrically homeomorphic to a Euclidean sphere endowed with a certain Finsler metric and vice versa. Based on these results, we present a classification of Finsler manifolds which admit a transnormal function. Specifically, we show that if a complete Finsler manifold admits a transnormal function with exactly two critical points, then it is homeomorphic to a sphere.
1. H. Akbar-Zadeh, Sur les espaces de Finsler ‘a courbures sectionnelles constantes. Acad. Roy. Belg. Bull. Cl. Sci. 74, 1 (1988), 281322.
2. H. Akbar-Zadeh, Initiation to Global Finsler Geometry. Elsevier, North Holland, 2006.
3. M. M. Alexandrino, B. O. Alves and H. R. Dehkordi, On Finsler transnormal functions. Differential. Geom. Appl. 65(2019), 93-107.
4. A. Asanjarani and B. Bidabad, Classification of complete Finsler manifolds through a second order differential equation, Differential Geom. Appl. 26(2008), 434-444.
5. D. Bao, S. S. Chern and Z. Shen, An introduction to Riemann-Finsler geometry. New York, Springer, 2012.
6. D. Bao and Z. Shen, Finsler metrics of constant positive curvature on the Lie group S 3,J. Lond. Math. Soc. 66(2002), 453-467.
7. B. Bidabad, On compact Finsler spaces of positive constant curvature, C. R. Math. Acad. Paris. 349(2011), 1191-1194.
8. B. Bidabad and A. Tayebi, A classification of some Finsler connections and their applications. Publ. Math. Debrecen. 71(2007), 253-266.
9. N. Boonnam, R. Hama and S. V. Sabau, Berwald spaces of bounded curvature are Riemannian, Acta Math. Acad. Paedagog. Nyiregyhaziensis (2017), 339-347.
10. A. Cauchy, Sur les polygones et les poly´edres: second m‘emoire. lEcole Polytechnique, XVIe Cahier. 9(1813), 26-38.
11. H. R. Dehkordi, Finsler Transnormal functions and singular foliations of codimension 1. PhD thesis, PhD thesis at IME University of Sao paulo, 2018.
12. H. R. Dehkordi, Mathematical modeling the wildfire propagation in a randers space, arXiv preprint arXiv:2012.06692 (2020).
13. H. R. Dehkordi, Mathematical modeling of wildfire propagation in an agricultural land, Proc. Ser. Braz. Soc. Comput. Appl. Math. 8(2021).
14. H. R. Dehkordi, Applications of Randers geodesics for wildfire spread modelling, Appl. Math. Model (2022).
15. H. R. Dehkordi and A. Saa, Huygens envelope principle in Finsler spaces and analogue gravity. Class. Quantum Grav. 36(2019), 085008.
16. S. Deng and M. Xu, Recent progress on homogeneous Finsler spaces with positive curvature, Eur. J. Math. 4(2017), 974-999.
17. C. Ekici and C. Muradiye, A note on Berwald eikonal equation, In J. Phys.: Conf. Ser. 766(2016), 012029.
18. P. Foulon, Locally symmetric Finsler spaces in negative curvature, Compt. Rendus. Acad. Sci. Math. 324(1997), 11271132.
19. P. Foulon, Curvature and global rigidity in Finsler manifolds. Houston J. Math. 28(2002), 263-292.
20. S. Gallot, Equations diff‘erentielles caract‘eristiques de la sphere, In Ann. Sci. de lEcole. Norm. Superieure. 12(1979), 235-267.
21. P. M. Gruber and J. M. Wills, Chapter 1.7 of Handbook of Convex Geometry, Vol. A. North-Holland Publishing Co., 1993.
22. T. Gudlaugur, Chapter 10 - a survey on isoparametric hypersurfaces and their generalizations. In Handbook of Differential Geometry, F. J. Dillen and L. C. Verstraelen, Eds., vol. 1 of Handbook of Differential Geometry. North-Holland, (2000), 963-995.
23. Q. He, S. Yin and Y. Shen, Isoparametric hypersurfaces in minkowski spaces, Differ. Geom. Appl. 47(2016), 133-158.
24. C. Kim and J. Yim, Rigidity of noncompact Finsler manifolds, Geom. Dedicata. 81(2000), 245-259.
25. C. Kim and J. Yim, Finsler manifolds with positive constant flag curvature, Geom. Dedicata. 98(2003), 47-56.
26. C. W. Kim, Locally symmetric positively curved Finsler spaces, Arch. Math. 88(2007), 378-384.
27. D. Lehmann, Une g´en´eralisation de la g´eom´etrie du plongement. S´eminaire Ehresmann, Topologie et g´eom´etrie diff´erentielle. 6(1964), 1-21.
28. H. B. Rademacher, A sphere theorem for non-reversible Finsler metrics, Math. Ann. 328(2004), 373-387.
29. Z. Shen, Lectures on Finsler geometry, World Scientific, Singapore, 2001.
30. Z. Shen, Finsler manifolds with nonpositive flag curvature and constant S-curvature, Math. Zeitschrift. 249(2005), 625-639.
31. Z. Shen, Differential geometry of spray and Finsler spaces, Springer, Netherlands, 2013.
32. Q. M. Wang, Isoparametric functions on Riemannian manifolds, Math. Ann. 277(1987), 639-646.
33. B. Wilking and W. Ziller, Revisiting homogeneous spaces with positive curvature, J. f¨ur die Reine und Angew. Math. 738(2018), 313-328.
34. B. Y. Wu, Some rigidity theorems for locally symmetrical Finsler manifolds, J. Geom. Phys. 58(2008), 923-930.
35. G. Wu and R. Ye, A note on Obata’s rigidity theorem, Commun. Math. Stat. 2(2014), 231-252.
36. M. Xu and J. A. Wolf, Sp(2)/U(1) and a positive curvature problem, Differ. Geom. Appl. 42(2015), 115-124.
37. M. Xu, L. Zhang, et al. δ-homogeneity in Finsler geometry and the positive curvature problem, Osaka J. Math. 55(2018), 177-194.
Asanjarani, A., & R. Dehkordi, H. (2022). Some rigidity results on complete Finsler manifolds. Journal of Finsler Geometry and its Applications, 3(1), 100-117. doi: 10.22098/jfga.2022.10415.1061
MLA
Azam Asanjarani; Hengameh R. Dehkordi. "Some rigidity results on complete Finsler manifolds", Journal of Finsler Geometry and its Applications, 3, 1, 2022, 100-117. doi: 10.22098/jfga.2022.10415.1061
HARVARD
Asanjarani, A., R. Dehkordi, H. (2022). 'Some rigidity results on complete Finsler manifolds', Journal of Finsler Geometry and its Applications, 3(1), pp. 100-117. doi: 10.22098/jfga.2022.10415.1061
VANCOUVER
Asanjarani, A., R. Dehkordi, H. Some rigidity results on complete Finsler manifolds. Journal of Finsler Geometry and its Applications, 2022; 3(1): 100-117. doi: 10.22098/jfga.2022.10415.1061