[Paper Review] Examples of flag-wise positively curved spaces

Ming Xu|arXiv (Cornell University)|Jun 6, 2016

Advanced Differential Geometry Research8 references3 citations

TL;DR

This paper constructs flag-wise positively curved Finsler metrics on compact Lie groups and coset spaces using a perturbation technique based on Killing navigation. It proves that any compact non-Abelian Lie group with trivial center admits such metrics, and more generally, that all compact simply connected coset spaces $G/H$ and their $S^1$-products admit flag-wise positively curved Finsler metrics, many of which are non-homogeneous.

ABSTRACT

A Finsler space $(M,F)$ is called flag-wise positively curved, if for any $x\in M$ and any tangent plane $\mathbf{P}\subset T_xM$, we can find a nonzero vector $y\in \mathbf{P}$, such that the flag curvature $K^F(x,y, \mathbf{P})>0$. 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 $\mathfrak{g}$ is compact non-Abelian and $\dim\mathfrak{c}(\mathfrak{g})\leq 1$ 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 simply connected coset space, $G/H$ and $S^1 imes G/H$ admit flag-wise positively curved Finsler metrics. This provides abundant examples for this type of metrics, which are not homogeneous in general.

Motivation & Objective

• To construct new examples of Finsler metrics satisfying the flag-wise positive curvature condition (FP), which is weaker than global positive curvature.
• To address Problem 4.4 from prior work by proving the existence of such metrics on compact non-Abelian Lie groups with small center.
• To extend the construction to general compact simply connected coset spaces $G/H$ and their $S^1$-products.
• To demonstrate that generic Finslerian perturbations of non-negatively curved homogeneous metrics can yield flag-wise positively curved metrics.
• To show that such metrics are abundant and not necessarily homogeneous, broadening the class of known examples.

Proposed method

• Uses the Killing navigation technique to perturb a bi-invariant or normal homogeneous Finsler metric on a compact Lie group.
• Applies a partition of unity on the unit sphere bundle $\mathcal{S}M$ to glue local Finsler metrics defined via navigation.
• Constructs local Finsler metrics $\tilde{F}_{i;\epsilon}$ on open sets $\mathcal{U}_i$ using Killing vector fields with small norm.
• Ensures the global metric $F_\epsilon = \sum \mu_i F_{i;\epsilon}$ remains positively definite via smooth partition of unity.
• Relies on the flag curvature formula $K^F(x,y,\mathbf{P}) = \frac{\langle R_y v, v \rangle_y^F}{\|y\|_y^2 \|v\|_y^2 - \langle y,v \rangle_y^2}$ to verify positivity.
• Uses topological arguments (e.g., contradiction via finite intersection of circles with co-dimension one spheres) to ensure every tangent plane intersects a region where flag curvature is positive.

Experimental results

Research questions

• RQ1 Can flag-wise positively curved Finsler metrics be constructed on compact non-Abelian Lie groups with $\dim \mathfrak{c}(\mathfrak{g}) \leq 1$?
• RQ2 Do all compact simply connected coset spaces $G/H$ admit flag-wise positively curved Finsler metrics?
• RQ3 Can such metrics be constructed without requiring homogeneity, thus expanding beyond known homogeneous examples?
• RQ4 Is a generic Finslerian perturbation of a non-negatively curved metric sufficient to achieve the flag-wise positive curvature condition?
• RQ5 Can the $S^1$-product construction preserve the flag-wise positive curvature property?

Key findings

• Any compact non-Abelian Lie group with $\dim \mathfrak{c}(\mathfrak{g}) \leq 1$ admits a left-invariant flag-wise positively curved Finsler metric.
• For any compact simply connected coset space $G/H$, both $G/H$ and $S^1 \times G/H$ admit flag-wise positively curved Finsler metrics.
• The constructed metrics are not necessarily homogeneous, showing that such metrics are abundant beyond the homogeneous class.
• The flag curvature remains positive for generic tangent vectors in every tangent plane, satisfying the (FP) condition globally.
• The construction relies on a carefully designed gluing process using Killing navigation and partition of unity on the sphere bundle.
• The method ensures the global metric remains Finsler (positive definite Hessian) while achieving flag-wise positive curvature through local perturbations.