Quarter-symmetric connection on an almost Hermitian manifold and on a Kähler manifold

Yıl 2024, Cilt: 53 Sayı: 4, 963 - 980, 27.08.2024

Milan Lj Zlatanovic , Miroslav Maksimovic

https://doi.org/10.15672/hujms.1219762

Öz

The paper observes an almost Hermitian manifold as an example of a generalized Riemannian manifold and examines the application of a quarter-symmetric connection on the almost Hermitian manifold. The almost Hermitian manifold with quarter-symmetric connection preserving the generalized Riemannian metric is actually the Kähler manifold. Observing the six linearly independent curvature tensors with respect to the quarter-symmetric connection, we construct tensors that do not depend on the quarter-symmetric connection generator. One of them coincides with the Weyl projective curvature tensor of symmetric metric $g$. Also, we obtain the relations between the Weyl projective curvature tensor and the holomorphically projective curvature tensor. Moreover, we examine the properties of curvature tensors when some tensors are hybrid.

Anahtar Kelimeler

almost Hermitian manifold, curvature tensors, hybrid tensor, Kähler manifold, quarter-symmetric connection, torsion tensor

Destekleyen Kurum

Ministry of Education, Science and Technological Development of the Republic of Serbia

Proje Numarası

451-03-9/2021-14/200124, 451-03-9/2021-14/200123

Kaynakça

• [1] S. Bhowmik, Some properties of a quarter-symmetric nonmetric connection in a Kähler manifold, Bull. Kerala Math. Assoc. 6 (1), 99–109, 2010.
• [2] S. Bulut, A quarter-symmetric metric connection on almost contact B-metric manifolds, Filomat 33 (16), 5181–5190, 2019.
• [3] B.B. Chaturvedi and B.K. Gupta, Study of a hyperbolic Kaehlerian manifolds equipped with a quarter-symmetric metric connection, Facta Universitatis, Ser. Math. Inform. 30 (1), 115–127, 2015.
• [4] B. B. Chaturvedi and P.N. Pandey, Kähler manifold with a special type of semisymmetric non-metric connection, Global Journal of Mathematical Sciences 7 (1), 17–24, 2015.
• [5] S. Chaubey and R. Ojha, On a semi-symmetric non-metric and quarter symmetric metric connections, Tensor N.S. 70, 202–213, 2008.
• [6] U.C. De, P. Zhao, K. Mandal and Y. Han, Certain curvature conditions on P-Sasakian manifolds admitting a quarter-symmetric metric connection, Chinese Ann. Math. Ser. B 41 (1), 133–146, 2020.
• [7] A.K. Dubey and R. H. Ojha, Some properties of quarter-symmetric non-metric connection in a Kähler manifold, Int. J. Contemp. Math. Sci. 5 (20), 1001–1007, 2010.
• [8] P. Gauduchon, Hermitian connections and Dirac operators, Unione Matematica Italiana, Bollettino B. 11, 257–288, 1997.
• [9] S. Golab, On semi-symmetric and quarter-symmetric linear connections, Tensor N.S. 29, 249–254, 1975.
• [10] Y. Han, H.T. Yun and P. Zhao, Some invariants of quarter-symmetric metric connections under projective transformation, Filomat 27 (4), 679–691, 2013.
• [11] S. Ivanov and M. Zlatanović, Connections on a non-symmetric (generalized) Riemannian manifold and gravity, Class. Quantum Grav. 33, 075016, 2016.
• [12] S. Ivanov and M. Zlatanović, Non-symmetric Riemannian gravity and Sasaki-Einstein 5-manifolds, Class. Quantum Grav. 37, 2020.
• [13] M.N.I. Khan, Tangent bundle endowed with quarter-symmetric non-metric connection on an almost Hermitian manifold, Facta Universitatis, Ser. Math. Inform. 35 (1), 167– 178, 2020.
• [14] J. Mikeš, E. Stepanova, A. Vanžurova, et al., Differential geometry of special mappings, Palacky University, Olomouc, 2015.
• [15] R.S. Mishra and S. Pandey, On quarter symmetric metric F-connections, Tensor N.S. 34, 1–7, 1980.
• [16] F. Özdemir and G.C. Yildirim, On conformally recurrent Kahlerian Weyl spaces, Topol. Appl. 153, 477–484, 2005.
• [17] M. Petrović, N. Vesić and M. Zlatanović, Curvature properties of metric and semisymmetric linear connections, Quaes. Math. 45 (10), 1603–1627, 2022.
• [18] M. Prvanović, Einstein connection of almost Hermitian manifold, Bulletin. Classe des Sciences Mathematiques et Naturelles. Sciences Mathematiques 20, 51–59, 1995.
• [19] M. Prvanović, Holomorphically projective curvature tensors, Kragujevac J. Math. 28, 97–111, 2005.
• [20] S.C. Rastogi, Some curvature properties of quarter symmetric metric connections, International Atomic Energy Agency (IAEA), International Centre for Theoretical Physics (ICTP) 18 (6), reference number 18015243, 1986.
• [21] W. Tang, T.Y. Ho, K.I. Ri, F. Fu and P. Zhao, On a generalized quarter-symmetric metric recurrent connection, Filomat 32 (1), 207–215, 2018.
• [22] M.M. Tripathi, A new connection in a Riemannian manifold, Int. Elec. J. Geom. 1 (1), 15–24, 2008.
• [23] V.V. Vishnevskii, A.P. Shirokov and V.V. Shurygin, Spaces over algebras (Prostranstva nad algebrami) (in Russian), Kazanskii Gosudarstvennyi Universitet, Kazan, 1985.
• [24] K. Yano, Differential geometry of complex and almost complex spaces, Pergamon Press, New York, 1965.
• [25] K. Yano, The Hayden connection and its applications, SEA Bull. Math. 6, 96–114, 1982.
• [26] K. Yano and T. Imai, Quarter-symmetric connections and their curvature tensors, Tensor N.S. 38, 13–18, 1982.
• [27] C. Yu, Curvature identities on almost Hermitian manifolds and applications, Sci. China Math. 60 (2), 285–300, 2016.
• [28] M. Zlatanović and M. Maksimović, Quarter-symmetric generalized metric connections on a generalized Riemannian manifold, Filomat 37 (12), 3927–3937, 2023.