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Inverse problem of left invariant sprays on Lie groups. (English) Zbl 1480.53030

In this interesting paper, the authors deal with the inverse problem in spray geometry. They find infinitely many sprays with non-diagonalizable Riemann curvature on a Lie group, where these sprays are not induced by Finsler metrics. Also, the left-invariant sprays with non-vanishing spray vectors on Lie groups are studied. It is proved that if such a spray \(S\) on a Lie group \(G\) satisfies that \(G\) is commutative or \(S\) is projective, then \(S\) is not induced by any (not necessary positive definite) left-invariant Finsler metric. Finally, an abundance of the left-invariant sprays on Lie groups are constructed, which satisfy the conditions in the above result.

MSC:

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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References:

[1] Bai, C. and Meng, D., Lie Group (Science Press, 2003) (in Chinese).
[2] Bao, D. and Shen, Z., Finsler metrics of constant flag curvature on the Lie group \(S^3\), J. London Math. Soc.66 (2002) 453-467. · Zbl 1032.53063
[3] Bucataru, I., Funk functions and projective deformations of sprays and Finsler spaces of scalar flag curvature, J. Geom. Anal.26 (2016) 3056-3065. · Zbl 1358.53079
[4] Bucataru, I. and Muzsnay, Z., Sprays metrizable by Finsler functions of constant flag curvature, Differential Geom. Appl.31 (2013) 405-415. · Zbl 1281.53074
[5] Huang, L., Einstein Finsler metrics on \(S^3\) with nonconstant flag curvature, Houston J. Math.37 (2011) 1071-1086. · Zbl 1235.53050
[6] Huang, L., Ricci curvature of left invariant Finsler metrics on Lie groups, Isr. J. Math.207 (2015) 783-792. · Zbl 1325.53097
[7] Li, Y., Mo, X. and Yu, Y., Inverse problem of sprays with scalar curvature, Internat. J. Math.30 (2019) 1950041. · Zbl 1422.53062
[8] Li, B. and Shen, Z., Sprays of isotropic curvature, Internat. J. Math.29 (2018) 1850003. · Zbl 1381.53144
[9] Muzsnay, Z., The Euler-Lagrange PDE and Finsler metrizability, Houston J. Math.32 (2006) 79-98. · Zbl 1113.53049
[10] Shen, Z., Differential Geometry of Spray and Finsler Spaces (Kluwer Academic Publishers, Dordrecht, 2001). · Zbl 1009.53004
[11] Yang, G., Some classes of sprays in projective spray geometry, Diff. Geom. Appl.29 (2011) 606-614. · Zbl 1219.53027
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