Abstract
We introduce the notions of spray vector and connection operator to give efficient curvature formulas for a homogeneous Finsler space. Thus the flag curvatures can be computed in the Lie algebra level. Applying these formulas, one can show that in several occasions the structure of the Lie algebra may have influence over the signs of the flag curvatures, regardless of the underlying Finsler metric. Some concrete examples are constructed to illustrate the concepts and the curvature behavior in Finsler geometry.
Similar content being viewed by others
References
Alekseevskii, D.V., Kimel’fel’d, B.N.: Structure of homogeneous Riemann spaces with zero Ricci curvature. Funct. Anal. Appl. 9(2), 97–102 (1975)
Aloff, S., Wallach, N.R.: An infinite family of 7-manifolds admitting positively curved Riemannian structures. Bull. Amer. Math. Soc. 81(1), 93–97 (1975)
Álvarez Paiva, J.C., Durán, C.E.: Isometric submersions of Finsler manifolds. Proc. Amer. Math. Soc. 129(8), 2409–2417 (2001)
Bao, D., Chern, S.-S., Shen, Z.: An Introduction to Riemann–Finsler Geometry. Graduate Texts in Mathematics, vol. 200. Springer, New York (2000)
Bao, D., Robles, C., Shen, Z.: Zermelo navigation on Riemannian manifolds. J. Differential Geom. 66(3), 377–435 (2004)
Bao, D., Shen, Z.: Finsler metrics of constant curvature on the Lie group \(S^3\). J. London Math. Soc. 66(2), 453–467 (2002)
Bérard-Bergery, L.: Les variétés riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positive. J. Math. Pures Appl. 55(1), 47–67 (1976)
Berger, M.: Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive. Ann. Sc. Norm. Super. Pisa 15, 179–246 (1961)
Besse, A.L.: Einstein Manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 10. Springer, Berlin (1987)
Chern, S.-S., Shen, Z.: Riemann–Finsler Geometry. Nankai Tracts in Mathematics, vol. 6. World Scientific, Singapore (2005)
Deng, S.: Fixed points of isometries of a Finsler space. Publ. Math. Debrecen 72(3–4), 469–474 (2008)
Deng, S., Hou, Z.: The group of isometries of a Finsler space. Pacific J. Math. 207(1), 149–155 (2002)
Deng, S., Hou, Z.: Invariant Finsler metrics on homogeneous manifolds. J. Phys. A 37(34), 8245–8253 (2004)
Deng, S., Hou, Z.: Naturally reductive homogeneous Finsler spaces. Manuscripta Math. 131(1–2), 215–229 (2010)
Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Pure and Applied Mathematics. vol. 80, 2nd edn. Academic Press, New York (1978)
Hu, Z., Deng, S.: Curvatures of homogeneous Randers spaces. Adv. Math. 240, 194–226 (2013)
Huang, L.: Einstein Finsler metrics on \(S^3\) with nonconstant flag curvature. Houston J. Math. 37(4), 1071–1086 (2011)
Huang, L.: On the fundamental equations of homogeneous Finsler spaces. Differential Geom. Appl. 40, 187–208 (2015)
Huang, L.: Ricci curvatures of left invariant Finsler metrics on Lie groups. Israel J. Math. 207(2), 783–792 (2015)
Jensen, G.R.: Einstein metrics on principal fibre bundles. J. Differential Geom. 8(4), 599–614 (1973)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Vols. 1, 2. Interscience, New York (1963, 1969)
Latifi, D.: Homogeneous geodesics in homogeneous Finsler spaces. J. Geom. Phys. 57(5), 1421–1433 (2007)
Lohkamp, J.: Metrics of negative Ricci curvature. Ann. Math. 140(3), 655–683 (1994)
Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21(3), 293–329 (1976)
Rademacher, H.-B.: A sphere theorem for non-reversible Finsler metrics. Math. Ann. 328(3), 373–387 (2004)
Shen, Z.: Projectively flat Finsler metrics of constant flag curvature. Trans. Amer. Math. Soc. 355(4), 1713–1728 (2003)
Shen, Z.: On projectively flat \((\alpha,\beta )\)-metrics. Canadian Math. Bull. 52(1), 132–144 (2009)
Shen, Z., Yu, C.: On a class of Einstein Finsler metrics. Internat. J. Math. 25(4), # 1450030 (2014)
Szabó, Z.I.: Positive definite Berwald spaces. Structure theorems on Berwald spaces. Tensor (NS) 35(1), 25–39 (1981)
Wallach, N.R.: Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. 96(2), 277–295 (1972)
Wolf, J.A.: Curvature in nilpotent Lie groups. Proc. Amer. Math. Soc. 15(2), 271–274 (1964)
Wolter, T.H.: Einstein metrics on solvable groups. Math. Z. 206(3), 457–471 (1991)
Wilking, B.: The normal homogeneous space \({\rm SU}(3){\times }{\rm SO}(3)/{\rm U}^\bullet (2)\) has positive sectional curvature. Proc. Amer. Math. Soc. 127(4), 1191–1994 (1999)
Wilking, B., Ziller, W.: Revisiting homogeneous spaces with positive curvature. J. Reine Angew. Math. doi:10.1515/crelle-2015-0053
Xu, M., Deng, S.: Normal homogeneous Finsler spaces. Transform. Groups. doi:10.1007/s00031-017-9428-7
Xu, M., Deng, S.: Towards the classification of odd-dimensional homogeneous reversible Finsler spaces with positive flag curvature. Ann. Mat. Pura Appl. doi:10.1007/s10231-016-0624-1
Xu, M., Deng, S., Huang, L., Hu, Z.: Even dimensional homogeneous Finsler spaces with positive flag curvature (2014). arXiv:1407.3582
Ziller, W.: Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259(3), 351–358 (1982)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by the National Natural Science Foundation of China Grant No. 11301283.
Rights and permissions
About this article
Cite this article
Huang, L. Flag curvatures of homogeneous Finsler spaces. European Journal of Mathematics 3, 1000–1029 (2017). https://doi.org/10.1007/s40879-017-0157-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40879-017-0157-1