# ON WEAKLY-BERWALD SPACES OF SPECIAL (α, β)-METRICS

• Published : 2006.05.01
• 46 5

#### Abstract

We have two concepts of Douglas spaces and Lands-berg spaces as generalizations of Berwald spaces. S. Bacso gave the definition of a weakly-Berwald space [2] as another generalization of Berwald spaces. In the present paper, we find the conditions that the Finsler space with an (${\alpha},{\beta}$)-metric be a weakly-Berwald space and the Finsler spaces with some special (${\alpha},{\beta}$)-metrics be weakly-Berwald spaces, respectively.

#### Keywords

Berwald space;cubic metric space;Douglas space;Finsler space with $L={\alpha}+{\beta}^2/{\alpha}$;infinite series (${\alpha},{\beta}$)-metric space;weakly-Berwald space

#### References

1. S. Bacso and M. Matsumoto, On Finsler spaces of Douglas type. A generalization of the notion of Berwald space, Publ. Math. Debrecen 51 (1997), no. 3-4, 385-406
2. S. Bacso and B. Szilagyi, On a weakly Berwald Finsler space of Kropina type, Math. Pannon. 13 (2002), no. 1, 91-95
3. M. Hashiguchi, S. Hojo, and M. Matsumoto, Landsberg spaces of dimension two with ($\alpha,\;\beta$)-metric, Tensor (N. S.) 57 (1996), no. 2, 145-153
4. M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kaiseisha Press, Otsu, Saikawa (1986)
5. M. Matsumoto, The Berwald connection of a Finsler space with an ($\alpha,\;\beta$)-metric, Tensor (N. S.) 50 (1991), no. 1, 18-21
6. M. Matsumoto, Finsler spaces with ($\alpha,\;\beta$)-metric of Douglas type, Tensor (N. S.) 60 (1998), no. 2, 123-134
7. I. Y. Lee and H. S. Park, Finsler spaces with infinite series ($\alpha,\;\beta$)-metric, J. Korean Math. Soc. 41 (2004), no. 3, 567-589 https://doi.org/10.4134/JKMS.2004.41.3.567
8. I. Y. Lee and D. G. Jun, On two-dimensional Landsberg space of a cubic Finsler space, East Asian Math. J. 19 (2003), no. 2, 305-316
9. R. Yoshikawa and K. Okubo, The conditions for some ($\alpha,\;\beta$)-metric spaces to be weakly-Berwald spaces, Proceedings of the 38-th Symposium on Finsler geometry, Nov. 12-15, (2003), 54-57
10. M. Matsumoto, Theory of Finsler spaces with ($\alpha,\;\beta$)-metric, Rep. Math. Phys. 31 (1992), 43-83 https://doi.org/10.1016/0034-4877(92)90005-L
11. S. Bacso and R. Yoshikawa, Weakly-Berwald spaces, Publ. Math. Debrecen 61 (2002), no. 2, 219-231
12. M. Matsumoto and S. Numata, On Finsler space with a cubic metric, Tensor (N. S.) 33 (1979), no. 2, 153-162

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