## Tbilisi Mathematical Journal

### On the theory of $4$-th root Finsler metrics

Akbar Tayebi

#### Abstract

In this paper, we consider exponential change of Finsler metrics. First, we find a condition under which the exponential change of a Finsler metric is projectively related to it. Then we restrict our attention to the $4$-th root metric. Let $F=\sqrt[4]{A}$ be an $4$-th root Finsler metric on an open subset $U\subset \mathbb{R}^n$ and ${\bar F}=e^{\beta/F}F$ be the exponential change of $F$. We show that ${\bar F}$ is locally projectively flat if and only if it is locally Minkowskian. Finally, we obtain necessary and sufficient condition under which ${\bar F}$ be locally dually flat.

#### Note

The author would like to thank the anonymous referees for their suggestions and comments which helped in improving the paper.

#### Article information

Source
Tbilisi Math. J., Volume 12, Issue 1 (2019), 83-92.

Dates
Accepted: 20 December 2018
First available in Project Euclid: 26 March 2019

https://projecteuclid.org/euclid.tbilisi/1553565628

Digital Object Identifier
doi:10.32513/tbilisi/1553565628

Mathematical Reviews number (MathSciNet)
MR3954221

#### Citation

Tayebi, Akbar. On the theory of $4$-th root Finsler metrics. Tbilisi Math. J. 12 (2019), no. 1, 83--92. doi:10.32513/tbilisi/1553565628. https://projecteuclid.org/euclid.tbilisi/1553565628

#### References

• S.-I. Amari and H. Nagaoka, Methods of Information Geometry, AMS Translation of Math. Monographs, Oxford University Press, 2000.
• G.S. Asanov, Finslerian Extension of General Relativity, Reidel, Dordrecht, 1984.
• V. Balan, Notable submanifolds in Berwald-Moór spaces, BSG Proc. 17, Geometry Balkan Press 2010, 21-30.
• V. Balan and S. Lebedev, On the Legendre transform and Hamiltonian formalism in Berwald-Moór geometry, Diff. Geom. Dyn. Syst. 12(2010), 4-11.
• V. Balan and N. Brinzei, Einstein equations for $(h,v)$-Berwald-Moór relativistic models, Balkan. J. Geom. Appl. 11(2)(2006), 20-26.
• G. Hamel, Uber die Geometrien, in denen die Geraden die Kürtzesten sind, Math. Ann. 57(1903), 231-264.
• M. Matsumoto and H. Shimada, On Finsler spaces with 1-form metric. II. Berwald-Moór's metric $L=\left( y^{1}y^{2}...y^{n}\right) ^{1/n}$, Tensor N. S. 32(1978), 275-278.
• A. Rapcsák, Über die bahntreuen Abbildungen metrisher Räume, Publ. Math. Debrecen, 8(1961), 285-290.
• Z. Shen, Riemann-Finsler geometry with applications to information geometry, Chin. Ann. Math. 27(2006), 73-94.
• C. Shibata, On invariant tensors of $\beta$-changes of Finsler metrics, J. Math. Kyoto Univ. 24(1984), 163-188.
• H. Shimada, On Finsler spaces with metric $L= \sqrt[m]{a_{i_{1}i_{2}...i_{m}}y^{i_{1}}y^{i_{2}}...y^{i_{m}}},$ Tensor, N.S. 33(1979), 365-372.
• D.G. Pavlov, Space-Time Structure, Algebra and Geometry, Collected papers, TETRU, 2006.
• D.G. Pavlov, Four-dimensional time, Hypercomplex Numbers in Geometry and Physics, 1(2004), 31-39.
• A. Tayebi, On generalized 4-th root metrics of isotropic scalar curvature, Mathematica Slovaca, 68(2018), 907-928.
• A. Tayebi and Izadian, On weakly Landsberg fourth root $(\alpha, \beta)$-metrics, Global Journal. Advanced. Research. Classical and Modern Geometries, 7(2018), 65-72.
• A. Tayebi and B. Najafi, On $m$-th root Finsler metrics, J. Geom. Phys. 61(2011), 1479-1484.
• A. Tayebi and B. Najafi, On $m$-th root metrics with special curvature properties, C. R. Acad. Sci. Paris, Ser. I. 349(2011), 691-693.
• A. Tayebi, A. Nankali and E. Peyghan, Some curvature properties of Cartan spaces with m-th root metrics, Lithuanian. Math. Journal, 54(1) (2014), 106-114.
• A. Tayebi, A. Nankali and E. Peyghan, Some properties of m-th root Finsler metrics, J. Contemporary. Math. Analysis, 49(4) (2014), 157-166.
• A. Tayebi, E. Peyghan and M. Shahbazi Nia, On generalized $m$-th root Finsler metrics, Linear. Algebra. Appl. 437(2012), 675-683.
• A. Tayebi, E. Peyghan and M. Shahbazi Nia, On Randers change of m-th root Finsler metrics, Inter. Elec. J. Geom, 8(2015), 14-20.
• A. Tayebi and M. Razgordani, Four families of projectively flat Finsler metrics with ${\bf K}=1$ and their non-Riemannian curvature properties, RACSAM, 112(2018), 1463-1485.
• A. Tayebi and M. Shahbazi Nia, On Matsumoto change of m-th root Finsler metrics, Publications De L'institut Mathematique, tome 101(115) (2017), 183-190.
• A. Tayebi and M. Shahbazi Nia, A new class of projectively flat Finsler metrics with constant flag curvature ${\bf K}=1$, Differ. Geom. Appl, 41(2015), 123-133.
• A. Tayebi, T. Tabatabaeifar and E. Peyghan, On Kropina-change of m-th root Finsler metrics, Ukrainian J. Math, 66(1) (2014), 1027-3190.