Abstract
In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) condition), which means that in each tangent plane, there exists a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset
spaces admitting non-negatively curved homogeneous Finsler metrics
satisfying the (FP) condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any
Lie group whose Lie algebra is a rank 2 non-Abelian compact Lie algebra
admits a left invariant Finsler metric satisfying the (FP) condition.
As by-products, we find
the first example of non-compact coset space
{S^{3}\times\mathbb{R}}
which admits homogeneous flag-wise positively curved Finsler metrics.
Moreover, we find some non-negatively curved Finsler metrics
on
{S^{2}\times S^{3}}
and
{S^{6}\times S^{7}}
which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on
{S^{3}\times S^{3}}
, shedding some light on the long standing general Hopf conjecture.
Homogeneous Finsler spaces and the flag-wise positively curved condition