Examples of flag-wise positively curved spaces

作者：Xu, Ming^*

来源：Differential Geometry and its Applications, 2017, 52: 42-50.

DOI：10.1016/j.difgeo.2017.03.015

• Finsler metric
• Flag curvature
• Flag-wise positively curved condition
• Left invariant metric
• Killing navigation

A Finger space (M, F) is called flag-wise positively curved, if for any x is an element of M and any tangent plane P subset of TxM, we can find a nonzero vector y is an element of P, such that the flag curvature K-F (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 S-1 x 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.

• 出版日期2017-6
• 单位天津师范大学