2- University of Mazandaran , rafie-rad@umz.ac.ir

An -dimensional Riemannian manifold is said to be

Likewise the Riemannian case, a Finslerian manifold is said to be

Thanks to the works of Bieberbach and Schoenflies, we apply an group theoretic approach to classify flat Randers manifolds. The key idea is that the isometry group of Randers manifold is a subgroup of the Euclidean group. This fact, may ease our approach to find and count discrete, co-compact and torsion frees subgroups of . First we find the Bieberbach subgroups and then, we count those that could form an isometry subgroup.

Here, flatness of a generic Finsler manifold is aimed to be defined so that it generalizes the flatness for Riemannian manifolds. The following result outcome in dimension 2 and 3:

To classify the flat Randers manifolds, we find out that the flat Randers manifolds are flat Riemannian manifolds. Besides, the isometry group of a Randers manifold is a subgroup of the isometry group . Our discussion also apply the following results:

- Every dimensional flat Randers manifold is itself a flat Riemannian manifolds.
- Every dimensional flat Randers manifold is orientable.
- The non-Riemannian properties for generic Finsler metrics may cause obstructions for a Finsler manifold to be falt.

The following conclusions were drawn from this research.

- In dimensions 2 and 3, the only connected and closed flat Randers manifolds are the tori , respectively.
- Every dimensional flat Randers manifold is itself a flat Riemannian manifolds../files/site1/files/62/11Abstract(1).pdf

Type of Study: Research Paper |
Subject:
alg

Received: 2017/12/4 | Revised: 2020/09/6 | Accepted: 2019/01/14 | Published: 2020/01/28 | ePublished: 2020/01/28

Received: 2017/12/4 | Revised: 2020/09/6 | Accepted: 2019/01/14 | Published: 2020/01/28 | ePublished: 2020/01/28

Rights and permissions | |

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. |