Examples of flag-wise positively curved spaces

Abstract

A Finsler space (M,F) is called flag-wise positively curved, if for any x∈M and any tangent plane P⊂TxM, we can find a nonzero vector y∈P, such that the flag curvature KF(x,y,P)>0. Though compact positively curved spaces are very rare in both Riemannian and Finsler geometry, flag-wise positively curved metrics should be easy to be found. A generic Finslerian perturbation for a non-negatively curved homogeneous metric may have a big chance to produce flag-wise positively curved metrics. This observation leads our discovery of these metrics on many compact manifolds. First we prove any Lie group G such that its Lie algebra g is compact non-Abelian and dim⁡c(g)≤1 admits flag-wise positively curved left invariant Finsler metrics. Similar techniques can be applied to our exploration for more general compact coset spaces. We will prove, whenever G/H is a compact coset space with a finite fundamental group, G/H and S1×G/H admit flag-wise positively curved Finsler metrics. This provides abundant examples for this type of metrics, which are not homogeneous in general. These examples implies a significant difference between the flag-wise positively curved condition and the positively curved condition, even though they are reduced to the same one in Riemannian geometry.