TY  - JOUR
T1  - Investigating the Interplay of Codazzi Couplings and Connections in Metallic-like Pseudo-Riemannian Manifolds
AU  - Aktaş, Buşra
AU  - Gezer, Aydın
AU  - Durmaz, Olgun
PY  - 2025
DA  - October
Y2  - 2025
DO  - 10.36890/iejg.1672080
JF  - International Electronic Journal of Geometry
JO  - Int. Electron. J. Geom.
PB  - Kazım İlarslan
WT  - DergiPark
SN  - 1307-5624
SP  - 208
EP  - 229
VL  - 18
IS  - 2
LA  - en
AB  - In this paper, we explore the characteristics of metallic-like pseudo-Riemannian manifolds that employ various linear connections. We establish a series of relationships regarding these connections, specifically examining the links between distinct types and their counterparts. Our findings reveal that a particular type of connection is associated with another if and only if its counterpart exhibits a corresponding relationship, and vice versa. Furthermore, we present significant equalities pertaining to the intrinsic properties of the manifolds, illustrating how these characteristics interact within a metallic-like context. We discuss the conditions required for a particular type of coupling, revealing significant connections between the properties of different associated connections. Additionally, we derive crucial equivalences that emphasise the relationship between torsion, statistical structures, and Codazzi coupling across various connections. Collectively, our results offer a cohesive framework that clarifies the geometric and algebraic foundations of metallic-like pseudo-Riemannian manifolds, enhancing our understanding of their structure and properties.
KW  - Metallic-like pseudo-Riemannian manifolds
KW  - conjugate connections
KW  - Codazzi coupleds
CR  - Altunbaş, M.: Metallic Riemannian structures on the tangent bundles of Riemannian manifolds with g-natural metrics. Int. Electron. J. Geom.
16(1), 95–103 (2023) https://doi.org/10.36890/iejg.1145729.
CR  - Amari, S.: Differential-geometrical methods in statistics. Lecture notes in statistics. vol. 28. Springer-Verlag, New York (1985).
CR  - Bejan, C. L., Crasmareanu, M.: Conjugate connections with respect to a quadratic endomorphism and duality. Filomat. 30(9), 2367–2374 (2016).
https://doi.org/10.2298/FIL1609367B
CR  - Blaga, A. M., Nannicini, A.: On the geometry of metallic pseudo-Riemannian structures. Riv. Math. Univ. Parma (N.S.). 11(1), 69–87 (2020).
CR  - Blaga, A. M., Nannicini, A.: On curvature tensors of Norden and metallic pseudo-Riemannian manifolds. Complex Manifolds. 6(1), 150–159
(2019). https://doi.org/10.1515/coma-2019-0008
CR  - Calin, O., Matsuzoe, H., Zhang, J.: Generalizations of conjugate connections. World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ,
26–34 (2009).
CR  - Durmaz, O., Gezer, A.: Conjugate connections and their applications on pure metallic metric geometries. Ricerche Mat. (2023).
https://doi.org/10.1007/s11587-023-00782-0
CR  - Erkan, E., Takano, K., Gulbahar, M.: Locally product-like statistical manifolds and their hypersurfaces. Int. Electron. J. Geom. 16(2), 435-450
(2023). HTTPS://DOI.ORG/10.36890/IEJG.1307467
CR  - Erken, I. K., Murathan, C. C., Yazla, A.: Almost cosympletic statistical manifolds. Quaest. Math. 43(2), 265-282 (2020).
https://doi.org/10.2989/16073606.2019.1576069
CR  - Fei, T., Zhang, J.: Interaction of Codazzi couplings with (para-)Kähler geometry. Results Math. 72(4), 2037–2056 (2017).
https://doi.org/10.1007/s00025-017-0711-7
CR  - Gezer, A., Karaman, C.: On metallic Riemannian structures. Turkish J. Math. 39(6), 954–962 (2015). https://doi.org/10.3906/mat-1504-50
CR  - Hretcanu, C. E., Crasmareanu, M.: Metallic structures on Riemannian manifolds. Rev. Un. Mat. Argentina. 54(2), 15–27 (2013).
CR  - S. Lauritzen, Statistical manifolds. In: Amari, S., Barndorff-Nielsen, O., Kass, R., Lauritzen, S., Rao, C.R. (eds.) Differential Geometry in
Statistical Inference, IMS Lecture Notes, vol. 10, pp. 163–216. Institute of Mathematical Statistics, Hayward (1987).
CR  - Manea, A.: Metallic-like structures and metallic-like maps. Turkish J. Math. 47(5), 1539–1549 (2023). https://doi.org/10.55730/1300-0098.3446
CR  - H. Nagaoka, S. Amari, Differential geometry of smooth families of probability distributions, Technical Report (METR) 82–7, Dept. of
Math. Eng. and Instr., Univ. of Tokyo (1982).
CR  - Norden, A.P.: Affinely connected spaces GRMFL. Moscow (1976) (in Russian).
CR  - Nomizu, K., Simon, U.: Notes on conjugate connections. World Scientific Publishing Co. Inc. River Edge. NJ. 152–173 (1992).
CR  - Stepanov, S. E., Stepanova, E. S., Shandra, I. G.: Conjugate connections on statistical manifolds. Izv. Vyssh. Uchebn. Zaved. Mat. 10, 90–98
(2007). https://doi.org/10.3103/S1066369X07100052
CR  - Takano, K.: Statistical manifolds with almost contact structures and its statistical submersions. J. Geom. 85 (1-2), 171-187 (2006).
https://doi.org/10.1007/s00022-006-0052-2
CR  - Takano, K.: Statistical manifolds with almost complex structures. Tensor, New Ser. 72(3), 225-231 (2010).
CR  - Vanzura, J.: Integrability conditions for polynomial structures. Kodai Math. Sem. Rep. 27 (1–2), 42–50 (1976).
https://doi.org/10.2996/kmj/1138847161
CR  - Vilcu, A.D., Vilcu, G.E.: Statistical manifolds with almost quaternionic structures and quaternionic Kähler-like statistical submersions. Entropy.
17(9), 6213-6228 (2015). https://doi.org/10.3390/e17096213
CR  - Vilcu, G.E.: Almost product structures on statistical manifolds and para-Kähler like statistical submersions. Bull. Sci. Math. 171, Paper No. 103018,
21 pp (2021). https://doi.org/10.1016/j.bulsci.2021.103018
UR  - https://doi.org/10.36890/iejg.1672080
L1  - https://dergipark.org.tr/en/download/article-file/4755010
ER  -