×

Reversible homogeneous Finsler metrics with positive flag curvature. (English) Zbl 1375.53068

Summary: In this work, we continue with the classification for positively curve homogeneous Finsler spaces \((G/H,F)\). With the assumption that the homogeneous space \(G/H\) is odd dimensional and the positively curved metric \(F\) is reversible, we only need to consider the most difficult case left, i.e. when the isotropy group \(H\) is regular in \(G\). Applying the fixed point set technique and the homogeneous flag curvature formulas, we show that the classification of odd dimensional positively curved reversible homogeneous Finsler spaces coincides with that of L. Bérard Bergery in Riemannian geometry except for five additional possible candidates, i.e. \(\mathrm{SU}(4)/\mathrm{SU}(2)_{(1,2)}\mathrm{S}^{1}_{(1,1,1,-3)}\), \(\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,1)}\), \(\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,3)}\), \(\mathrm{Sp}(3)/\mathrm{Sp}(1)_{(3)}\mathrm{S}^{1}_{(1,1,0)}\), and \(G_{2}/\mathrm{SU}(2)\) with \(\mathrm{SU}(2)\) the normal subgroup of \(\mathrm{SO}(4)\) corresponding to the long root. Applying this classification to homogeneous positively curved reversible \((\alpha,\beta)\) metrics, the number of exceptional candidates can be reduced to only two, i.e. \(\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,1)}\) and \(\mathrm{Sp}(3)/\mathrm{Sp}(1)_{(3)}\mathrm{S}^{1}_{(1,1,0)}\).

MSC:

53C30 Differential geometry of homogeneous manifolds
22E46 Semisimple Lie groups and their representations
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] [1] S. Aloff and N. Wallach, An infinite family of 7-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975), 93-97. AloffS.WallachN.An infinite family of 7-manifolds admitting positively curved Riemannian structuresBull. Amer. Math. Soc.8119759397 · Zbl 0362.53033
[2] [2] D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemannian Finsler Geometry, Grad. Texts in Math. 200, Springer, New York, 2000. BaoD.ChernS. S.ShenZ.An Introduction to Riemannian Finsler GeometryGrad. Texts in Math. 200SpringerNew York2000 · Zbl 0954.53001
[3] [3] L. Bérard Bergery, Les variétés Riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positive, J. Math. Pures Appl. (9) 55 (1976), 47-68. Bérard BergeryL.Les variétés Riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positiveJ. Math. Pures Appl. (9)5519764768 · Zbl 0289.53037
[4] [4] M. Berger, Les varietes Riemanniennes homogenes normales simplement connexes a courbure strictment positive, Ann. Sc. Norm. Super. Pisa Sci. Fis. Mat. III. Ser. 15 (1961), 191-240. BergerM.Les varietes Riemanniennes homogenes normales simplement connexes a courbure strictment positiveAnn. Sc. Norm. Super. Pisa Sci. Fis. Mat. III. Ser.151961191240 · Zbl 0101.14201
[5] [5] S. Deng, Homogeneous Finsler Spaces, Springer, New York, 2012. DengS.Homogeneous Finsler SpacesSpringerNew York2012 · Zbl 1253.53002
[6] [6] S. Deng and M. Xu, Left invariant Clifford-Wolf homogeneous (α,β){(\alpha,\beta)}-metrics on compact semisimple Lie groups, Transform. Groups 20 (2015), no. 2, 395-416. DengS.XuM.Left invariant Clifford-Wolf homogeneous (α,β){(\alpha,\beta)}-metrics on compact semisimple Lie groupsTransform. Groups2020152395416 · Zbl 1320.53090
[7] [7] L. Huang, On the fundamental equations of homogeneous Finsler spaces, Differential Geom. Appl. 40 (2015), 187-208. HuangL.On the fundamental equations of homogeneous Finsler spacesDifferential Geom. Appl.402015187208 · Zbl 1320.53027
[8] [8] L. Kozma, Weinstein’s theorem for Finsler manifolds, Kyoto J. Math. 46 (2006), 377-382. KozmaL.Weinstein’s theorem for Finsler manifoldsKyoto J. Math.462006377382 · Zbl 1178.53077
[9] [9] N. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. (2) 96 (1972), 277-295. WallachN.Compact homogeneous Riemannian manifolds with strictly positive curvatureAnn. of Math. (2)961972277295 · Zbl 0261.53033
[10] [10] B. Wilking and W. Ziller, Revisiting homogeneous spaces with positive curvature, J. Reine Angew. Math. (2015), 10.1515/crelle-2015-0053. WilkingB.ZillerW.Revisiting homogeneous spaces with positive curvatureJ. Reine Angew. Math.201510.1515/crelle-2015-0053 · Zbl 1405.53076 · doi:10.1515/crelle-2015-0053
[11] [11] J. Wolf, Spaces of Constant Curvature, 5th ed., Publish or Perish, Boston, 1984. WolfJ.Spaces of Constant Curvature5th ed.Publish or PerishBoston1984 · Zbl 0556.53033
[12] [12] M. Xu and S. Deng, Normal homogeneous Finsler spaces, preprint (2014), http://arxiv.org/abs/1411.3053; to appear in Transform. Groups. XuM.DengS.Normal homogeneous Finsler spaces2014 <ext-link ext-link-type=”uri“ xlink.href=”>http://arxiv.org/abs/1411.3053; to appear in Transform. Groups
[13] [13] M. Xu and S. Deng, Towards the classication of odd dimensional homogeneous reversible Finsler spaces with positive flag curvature, preprint (2015), http://arxiv.org/abs/1504.03018. XuM.DengS.Towards the classication of odd dimensional homogeneous reversible Finsler spaces with positive flag curvature2015 <ext-link ext-link-type=”uri“ xlink.href=”>http://arxiv.org/abs/1504.03018
[14] [14] M. Xu and S. Deng, Homogeneous Finsler spaces and the flag-wise positively curved condition, preprint (2016), http://arxiv.org/abs/1604.07695. XuM.DengS.Homogeneous Finsler spaces and the flag-wise positively curved condition2016 <ext-link ext-link-type=”uri“ xlink.href=”>http://arxiv.org/abs/1604.07695
[15] [15] M. Xu, S. Deng, L. Huang and Z. Hu, Even dimensional homogeneous Finsler spaces with positive flag curvature, preprint (2014), http://arxiv.org/abs/1407.3582; to appear in Indiana Univ. Math. J. XuM.DengS.HuangL.HuZ.Even dimensional homogeneous Finsler spaces with positive flag curvature2014 <ext-link ext-link-type=”uri“ xlink.href=”>http://arxiv.org/abs/1407.3582; to appear in Indiana Univ. Math. J.
[16] [16] M. Xu and J. Wolf, Sp⁢(2)/U⁢(1){Sp(2)/U(1)} and a positive curvature problem, Differential Geom. Appl. 42 (2015), 115-124. XuM.WolfJ.Sp⁢(2)/U⁢(1){Sp(2)/U(1)} and a positive curvature problemDifferential Geom. Appl.422015115124 · Zbl 1326.53066
[17] [17] W. Ziller, Examples of Riemannian manifolds with nonnegative sectional curvature, Metric and Comparison Geometry, Surv. Differ. Geom. 11, International Pres, Somerville (2007), 63-102. ZillerW.Examples of Riemannian manifolds with nonnegative sectional curvatureMetric and Comparison GeometrySurv. Differ. Geom. 11International PresSomerville200763102 · Zbl 1153.53033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.