Geodesic
Homogeneous Einstein Finsler Metrics on $(4n+3)$-dimensional Spheres
Canadian mathematical bulletin, Tome 62 (2019) no. 3, pp. 509-523

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper, we study a class of homogeneous Finsler metrics of vanishing $S$-curvature on a $(4n+3)$-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature $1$ in this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on $S^{4n+3}$ of constant Ricci curvature $1$ and vanishing $S$-curvature. They contain the canonical metric on $S^{4n+3}$ of constant sectional curvature $1$ and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.
DOI : 10.4153/S0008439518000139
Mots-clés : Finsler metric, homogeneous space, Einstein metric 
Huang, Libing; Mo, Xiaohuan. Homogeneous Einstein Finsler Metrics on $(4n+3)$-dimensional Spheres. Canadian mathematical bulletin, Tome 62 (2019) no. 3, pp. 509-523. doi: 10.4153/S0008439518000139
@article{10_4153_S0008439518000139,
     author = {Huang, Libing and Mo, Xiaohuan},
     title = {Homogeneous {Einstein} {Finsler} {Metrics} on $(4n+3)$-dimensional {Spheres}},
     journal = {Canadian mathematical bulletin},
     pages = {509--523},
     year = {2019},
     volume = {62},
     number = {3},
     doi = {10.4153/S0008439518000139},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439518000139/}
}
TY  - JOUR
AU  - Huang, Libing
AU  - Mo, Xiaohuan
TI  - Homogeneous Einstein Finsler Metrics on $(4n+3)$-dimensional Spheres
JO  - Canadian mathematical bulletin
PY  - 2019
SP  - 509
EP  - 523
VL  - 62
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/S0008439518000139/
DO  - 10.4153/S0008439518000139
ID  - 10_4153_S0008439518000139
ER  - 
%0 Journal Article
%A Huang, Libing
%A Mo, Xiaohuan
%T Homogeneous Einstein Finsler Metrics on $(4n+3)$-dimensional Spheres
%J Canadian mathematical bulletin
%D 2019
%P 509-523
%V 62
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/S0008439518000139/
%R 10.4153/S0008439518000139
%F 10_4153_S0008439518000139

[1] Bao, D. and Robles, C., Ricci and flag curvature . In: A Sampler of Riemann-Finsler geometry , Math. Sci. Res. Inst. Publ., 50, Cambridge University Press, 2004. Google Scholar

[2] Cheng, X., Shen, Z., and Tian, Y., A class of Einstein (𝛼, 𝛽)-metrics . Israel J. Math. 192(2012), 221–249. . Google Scholar | DOI

[3] Chern, S. S., Finsler geometry is just Riemannian geometry without the quadratic restriction . Notices Amer. Math. Soc. 43(1996), 959–963. Google Scholar

[4] Deng, S. and Hou, Z., Invariant Finsler metrics on homogeneous manifolds . J. Phys. A 37(2004), n. 34, 8245–8253. . Google Scholar | DOI

[5] Huang, L., Einstein Finsler metrics on S 3 with nonconstant flag curvature . Houston J. Math. 37(2011), 1071–1086. Google Scholar

[6] Huang, L., Ricci curvature of left invariant Finsler metrics on Lie groups . Israel J. Math. 207(2015), 783–792. . Google Scholar | DOI

[7] Huang, L., On the fundamental equations of homogeneous Finsler spaces . Diff. Geom. Appl. 40(2015), 187–208. . Google Scholar | DOI

[8] Huang, L., Flag curvatures of homogeneous Finsler spaces . European J. Math. 3(2017), 1000–1029. . Google Scholar | DOI

[9] Jensen, G. R., Einstein metrics on principle fibre bundles . J. Diff. Geom. 8(1973), 599–614. Google Scholar

[10] Milnor, J., Curvatures of left invariant metrics on Lie groups . Adv. Math. 21(1976), 293–329. . Google Scholar | DOI

[11] Mo, X., On the flag curvature of a Finsler space with constant S-curvature . Houston J. Math. 31(2005), 131–144. Google Scholar

[12] Sevim, E. S., Shen, Z., and Zhao, L., On a class of Ricci flat Douglas metrics . Intern. J. Math. 23(2012), 1250046, 15pp. . Google Scholar | DOI

[13] Shen, Z., Volume comparison and its application in Riemann-Finsler geometry . Adv. Math. 128(1997), 306–328. . Google Scholar | DOI

[14] Shen, Z., Lectures on Finsler geometry . World Scientific Publishing Co., Singapore, 2001. . Google Scholar | DOI

[15] Shen, Z. and Yu, C., On a class of Einstein Finsler metrics . Internat. J. Math. 25(2014), 1450030, 18pp. . Google Scholar | DOI

[16] Wang, H., Huang, L., and Deng, S., Homogeneous Einstein-Randers metrics on spheres . Nonlinear Anal. 74(2011), 6295–6301. . Google Scholar | DOI

[17] Ziller, W., Homogeneous Einstein metrics on spheres and projective spaces . Math. Ann. 259(1982), 351–358. . Google Scholar | DOI

Cité par Sources :