Tripathi connection in Finsler geometry

Yıl 2023, Cilt: 52 Sayı: 1, 136 - 150, 15.02.2023

Amr Soleiman Ebtsam Taha

https://doi.org/10.15672/hujms.1081955

Öz

Adopting the pullback formalism, a new linear connection in Finsler geometry has been introduced and investigated. Such connection unifies all formerly known Finsler connections and some other connections not introduced so far. Also, our connection is a Finslerian version of the Tripathi connection introduced in Riemannian geometry. The existence and uniqueness of such connection is proved intrinsically. An explicit intrinsic expression relating this connection to Cartan connection is obtained. Some generalized Finsler connections are constructed from Tripathi Finsler connection, by applying the ${P}^{1}$-process and ${C}$-process introduced by Matsumoto. Finally, under certain conditions, many special Finsler connections are given.

Anahtar Kelimeler

spray, Finsler manifold, Cartan connection, horizontal recurrent connection, semi-symmetric connection, quarter-symmetric connection

Kaynakça

• [1] N. S. Agashe and M. R. Chafle, A semi-symmetric non-metric connection in a Riemannian manifolds, Indian J. Pure Appl. Math. 23, 399-409, 1992.
• [2] O. C. Andonie and D. Smaranda, Certaines connexions semi-symètriques, Tensor, N. S. 31, 8-12, 1977.
• [3] G. B. Folland, Weyl manifolds, J. Diff. Geom. 4, 143-153, 1970.
• [4] J. Grifone, Structure présque-tangente et connexions, I, Ann. Inst. Fourier, Grenoble, 22 (1), 287-334, 1972.
• [5] J. Grifone, Structure presque-tangente et connexions, II, Ann. Inst. Fourier, Grenoble, 22 (3), 291-338, 1972.
• [6] H. A. Hayden, Subspaces of a space with torsion, Proc. London Math. Soc. 34, 27-50, 1932.
• [7] J. Klein and A. Voutier, Formes extérieures génératrices de sprays, Ann. Inst. Fourier, Grenoble, 18 (1), 241-260, 1968.
• [8] Y. Liang, On semi-symmetric recurrent-metric connection, Tensor, N. S. 55, 107-112, 1994.
• [9] M. Matsumoto, The theory of Finsler connections, Publication of the study group of geometry, 5, Dept. Math., Okayama Univ. 1970.
• [10] M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kaiseisha Press, Japan, 1986.
• [11] R. Miron and M. Anastasiei, The geometry of Lagrange spaces: Theory and applications, Kluwer Acad. Publ. 1994.
• [12] R. S. Mishra and S. N. Pandey, On quarter-symmetric metric F-connections, Tensor, N. S. 34, 1-7, 1980.
• [13] B. N. Prasad and L. Srivastava, On hv-recurrent Finsler connection, Indian J. Pure Appl. Math. 20, 790-798, 1989.
• [14] H. Rund, The differential geometry of Finsler spaces, Springer-Verlag, Berlin, 1959.
• [15] J. Sengupta, U. C. De and T. Q. Binh, On a type of semi-symmetric non-metric connection on Riemannian manifolds, Indian J. Pure Appl. Math. 31, 659-1670, 2000.
• [16] A. Soleiman, S. G. Elgendi and A. M. Abdelsalam, A new general Finsler connection, J. Finsler Geom. Appl. (JFGA), 1, 1-14, 2020.
• [17] J. Szilasi, R. L. Lovas and D. Cs Kertesz, Connections, Sprays and Finsler structures, World Scientific, 2014.
• [18] W. Tang, T. Ho, K. Ri, F. Fu and P. Zhao, On a generalized quarter symmetric metric recurrent connection, Filomat 32 (1) , 207-215, 2018.
• [19] M. M. Tripathi, A new connection in a Riemannian manifold, Int. Electronic J. Geom. 1, 15-24, 2008.
• [20] Cs. Vincze, On a special type of generalized Berwald manifolds: semi-symmetric linear connections preserving theFinslerian length of tangent vectors, European J. Math. 3, 1098-1171, 2017.
• [21] K. Yano, On semi-symmetric metric connections, Rev. Roumaine Math. Pures Appl. 15, 1579-1586, 1970.
• [22] K. Yano, Integral formulas in Riemannian geometry, Marcel Dekker Inc. 1970.
• [23] K. Yano and T. Imai, Quarter-symmetric metric connections and their curvature tensors, Tensor, N. S. 38, 13-18, 1984.
• [24] Nabil L. Youssef, Connexions métiques semi-symétriques semi-basiques, Tensor, N. S. 40, 242-248, 1983.
• [25] Nabil L. Youssef, Vertical semi-symmetric metric connections Tensor, N. S., 49, 218- 229, 1990.
• [26] Nabil L. Youssef and A. Soleiman, On horizontal recurrent Finsler connection, Rend. Circ. Mat. Palermo, II. Ser, 68, 1-9, 2019.
• [27] Nabil L. Youssef, S. H. Abed and A. Soleiman, Cartan and Berwald connections in the pullback formalism, Algebras, Groups and Geometries, 25, 363-384, 2008.
• [28] Nabil L. Youssef, S. H. Abed and A. Soleiman, A global approach to the theory of special Finsler manifolds, J. Math. Kyoto Univ. 48, 857-893, 2008.
• [29] Nabil L. Youssef, S. H. Abed and A. Soleiman, A global approach to the theory of connections in Finsler geometry, Tensor, N. S. 71, 187-208, 2009.
• [30] Nabil L. Youssef, S. H. Abed and A. Soleiman, Geometric objects associated with the fundamental connections in Finsler geometry, J. Egypt. Math. Soc. 18, 67-90, 2010.