It is proved that every locally flat Finsler manifold is a locally flat Riemannian manifold. Some low dimensional locally Finsler manifolds are classified. It is also proved that in a categorical sense, there is a correspondence between locally flat Finsler manifolds and locally hessian Riemannian manifolds
1. L. Bieberbach, Uber die Bewegungsgruppen der Euklidischen R¨aume I ¨ , Math. Annalen, 70(3) (1911), 297-336.
2. L. Bieberbach, Uber die Bewegungsgruppen der Euklidischen R¨aume II: Die Gruppen ¨ mit einem endlichen Fundamentalbereich, Math. Annalen, 72 (3) (1912), 400-412.
3. X. Chen, X. Mo and Z. Shen, On the flag curvature of Finsler metrics of scalar curvature, J. London Math. Soc., 68(2003), 762-780.
4. A. Deicke, Uber die Finsler-R¨aume mit ¨ Ai = 0, Arch. Math. 4 (1953), 45-51.
5. S. Deng and Z. Hou, The Group of Isometries of a Finsler space, Pacific J. Math, 207(1), 2002.
6. N. Bogdan, The Mazur-Ulam theorem, arXiv: 1306.2380v1, (2013).
7. A. Schoenflies, Kristallsysteme und Kristallstruktur, Teubner (1891).
8. L. Charlap, Bieberbach Groups and Flat Manifolds, Springer, (1986).
9. S. B. Myers and N. E. Steenrod, The Group of Isometries of a Riemannian Manifold, Annals of Math. Second Series, 40 (2) (1939), 400-416.
10. Li. Chi-Kwong Norms, Isometries, and Isometry Groups, J. Amer. Math. Monthly., 107(4) (2000), 334-340.
11. J. Fennel, Isometries between finite-dimensional normed vector spaces, Preprint, (2016).
12. H. Shima, On certain locally flat homogeneous manifolds of solvable Lie groups, Osaka J. Math. 13 (1976), 213-229.
13. H. Shima, Compact locally hessian manifolds, Osaka J. Math. 15 (1978), 509-513.
14. A. Szczepa´nski, Geometry of crystallographic groups, Singapore, Singapor, World Scientific Publshing, (2012).
15. Z, Shen, Differential Geometry of Spray and Finsler spaces, Kluwer Academic Publishers, 2001.
16. J. Wolf, Spaces of constant curvature, New York, MacGrow Hill, (1967).
17. R.G. Torrom´e, Averaged structures associated to a Finsler structure, arXiv:math/0501058v9.
18. H.C. Wang, On Finsler spaces with completly integrable equations of Killing, J. Lond. Math. Soc. 22 (1947), 5-9.
Alavi, S. S. (2020). On the locally flat Finsler manifolds. Journal of Finsler Geometry and its Applications, 1(2), 115-129. doi: 10.22098/jfga.2020.1245
MLA
Seiedeh Sedigheh Alavi. "On the locally flat Finsler manifolds", Journal of Finsler Geometry and its Applications, 1, 2, 2020, 115-129. doi: 10.22098/jfga.2020.1245
HARVARD
Alavi, S. S. (2020). 'On the locally flat Finsler manifolds', Journal of Finsler Geometry and its Applications, 1(2), pp. 115-129. doi: 10.22098/jfga.2020.1245
VANCOUVER
Alavi, S. S. On the locally flat Finsler manifolds. Journal of Finsler Geometry and its Applications, 2020; 1(2): 115-129. doi: 10.22098/jfga.2020.1245