Low Dimensional Flat Manifolds with Some Elasses of Finsler Metric

An -dimensional Riemannian manifold  is said to be flat (or locally Euclidean) if  locally isometric with the Euclidean space, that is,  admits a covering of coordinates neighborhoods each of which is isometric with a Euclidean domain.  A Riemannian manifold  is flat if and only if  admits a covering of coordinates neighborhoods on each of, the function  is independent of . A classical result affirms that a Riemannian manifold is flat if and only if its Riemann curvature vanishes (equivalently, the sectional curvature; This is usually taken as the definition of a flat Riemannian manifold in the contexts. The universal Riemannian covering space of a complete and flat Riemannian manifold is the Euclidean space . Up to local isometry, Bieberbach proved that any compact flat Riemannian manifold, is realized as a quotient space , where  is a discrete, co-compact and torsion free subgroup of the Euclidean group  , cf.  The only 1 dimensional complete, flat and connected manifolds are  and . In 2 dimensions,  the only complete, flat and connected manifolds are cylinder, Mӧbius strip, Torus and Klein bottle. In 3 dimensions, there are only 10 complete, flat and connected manifolds including 6 oriented and 4 non-oriented manifolds, cf.  .Likewise the Riemannian case, a Finslerian manifold   is said to be flat (or locally Minkowskian) if,  admits a covering of coordinates neighborhoods each of which isometric with a single Minkowski normed domain. A Finslerian  manifold  is flat if and only if it admits a covering of coordinates neighborhoods on each of, the function  is independent of .  The flag curvature  of any flat Finsler manifold vanishes identically.

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:Theorem 1. The only connected and closed -dimensional () closed flat Randers manifolds is the torus  , respectively.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.