Every Finsler metric naturally induces a spray but not so for the converse. The notion sprays of scalar (resp. isotropic) curvature has been known as generalization metrics flag curvature. In this paper, new notion, constant curvature, is introduced and especially it shows that isotropic necessarily even in dimension $$n\ge 3$$ . Further, complete conditions are given constant) to be metrizable...

Among the compact homogeneous spaces, a very distinguished subclass is formed by the (generalized) real flag manifolds which by definition are the orbits of the isotropy representations of Riemannian symmetric spaces (sorbits). This class contains most compact symmetric spaces (e.g. all hermitian ones), all classical flag manifolds over real, complex and quaternionic vector spaces, all adjoint ...

In this paper, we prove that for every Finsler n-sphere (S, F ) for n ≥ 3 with reversibility λ and flag curvature K satisfying ( λ λ+1 )2 < K ≤ 1, either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form exp(πiμ) with an irrational μ. Furthermore, there always exist three...

In this paper, we prove that on every Finsler n-sphere (S, F ) for n ≥ 6 with reversibility λ and flag curvature K satisfying ( λ λ+1 )2 < K ≤ 1, either there exist infinitely many prime closed geodesics or there exist [ 2 ] − 2 closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist n − 3 closed geodesics possessing irrational average indice...