## Osaka Journal of Mathematics

### $\delta$-homogeneity in Finsler geometry and the positive curvature problem

#### Abstract

In this paper, we explore the similarity between normal homogeneity and $\delta$-homogeneity in Finsler geometry. They are both non-negatively curved Finsler spaces. We show that any connected $\delta$-homogeneous Finsler space is $G$-$\delta$-homogeneous, for some suitably chosen connected quasi-compact $G$. So $\delta$-homogeneous Finsler metrics can be defined by a bi-invariant singular metric on $G$ and submersion, just as normal homogeneous metrics, using a bi-invariant Finsler metric on $G$ instead. More careful analysis shows, in the space of all Finsler metrics on $G/H$, the subset of all $G$-$\delta$-homogeneous ones is in fact the closure for the subset of all $G$-normal ones, in the local $C^0$-topology (Theorem 1.1). Using this approximation technique, the classification work for positively curved normal homogeneous Finsler spaces can be applied to classify positively curved $\delta$-homogeneous Finsler spaces, which provides the same classification list. As a by-product, this argument tells more about $\delta$-homogeneous Finsler metrics satisfying the (FP) condition (a weaker version of positively curved condition).

#### Article information

Source
Osaka J. Math., Volume 55, Number 1 (2018), 177-194.

Dates
First available in Project Euclid: 11 January 2018

https://projecteuclid.org/euclid.ojm/1515661220

Mathematical Reviews number (MathSciNet)
MR3744979

Zentralblatt MATH identifier
06848747

#### Citation

Xu, Ming; Zhang*, Lei. $\delta$-homogeneity in Finsler geometry and the positive curvature problem. Osaka J. Math. 55 (2018), no. 1, 177--194. https://projecteuclid.org/euclid.ojm/1515661220

#### References

• D. Bao, S.S. Chern and Z. Shen: An Introduction to Riemannian-Finsler Geometry, G.T.M. 200, Springer, Berlin-Heidelberg-New York, 2000.
• M. Berger: Les variétés riemanniennes homogénes normales simplement connexes à courbure strictement positive, Ann. Scula Norm. Sup. Pisa (3) 15 (1961), 179–246.
• M. Berger: Trois remarques sur les variétés riemanniennes à courbure positive, C. R. Acad. Sci. Paris, Ser. A-B, 263 (1966), 76–78.
• V. Berestovskii and C. Plaut: Homogeneous spaces of curvature bounded below, J. Geom. Anal. 9 (1999), 203–219.
• V. Berestovskii and Y. Nikonorov: On $\delta$-homogeneous Riemannian manifolds, Differential Geom. Appl. 26 (2008), 514–535.
• V. Berestovskii and Y. Nikonorov: The Chebyshev norm on the Lie algebra of the motion group of a compact homogeneous manifold, J. Math Sci. 161 (2009), 97–121.
• V. Berestovskii and Y. Nikonorov: Generalized normal homogeneous Riemannian metrics on spheres and projective spaces, Ann. Global Anal. Geom. 45 (2014), 167–196.
• V. Berestovskii, E.V. Nikitenko and Y. Nikonorov: Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic, Differential Geom. Appl. 29 (2011), 533–546.
• S. Deng: Homogeneous Finsler spaces, Springer, Berlin-Heidelberg-New York, 2012.
• S. Deng and M. Xu: Clifford-Wolf translations of Finsler spaces, Forum Math. 26 (2014), 1413–1428.
• L. Huang: Ricci curvature of left invariant Finsler metrics on Lie groups, Israel J. Math. 207 (2015), 783–792.
• J.C. Alvarez Paive and C.E. Duran: Isometric submersion of Finsler manifolds, Proc. Amer. Math. Soc. 129 (2001), 2409–2417.
• Z. Shen: Lectures on Finsler Geometry, World Scientific, 2001.
• N.R. Wallach: Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. (2) 96 (1972), 277–295.
• B. Wilking: The normal homogeneous space $(\mathrm{SU}(3)\times \mathrm{SO(3)})/\mathrm{U}^*(2)$ has positive sectional curvature, Proc. Amer. Math. Soc. 127 (1999), 1191–1194.
• M. Xu: Examples of flag-wise positively curved spaces, preprint, arXiv:1606.01731.
• M. Xu and S. Deng: Normal homogeneous Finsler spaces, Transform. Groups (to appear), arXiv:1411.3053.
• M. Xu and S. Deng: Homogeneous Finsler spaces and the flag-wise positively curved condition, preprint, arXiv:1604.07695.
• M. Xu, S. Deng, L. Huang and Z. Hu: Even dimensional homogeneous Finsler spaces with positive flag curvature, Indiana Univ. Math. J. (to appear), arXiv:1407.3582.
• L. Zhang and S. Deng: On generalized normal homogeneous Randers spaces, Publ. Math. Debrecen (to appear).