Extending Information Geometry: Quasi-Statistical Structures on Metallic-Like Pseudo-Riemannian Manifolds

Aydın Gezer; Olgun Durmaz; Buşra Aktaş

doi:10.47000/tjmcs.1789376

Extending Information Geometry: Quasi-Statistical Structures on Metallic-Like Pseudo-Riemannian Manifolds

Abstract

In this paper, we investigate the conditions under which quasi-statistical structures can be realized on metallic-like pseudo-Riemannian manifolds. By combining the flexibility of quasi-statistical geometry with the algebraic richness of metallic-like structures, we provide a unified framework for analyzing compatibility conditions among metrics, conjugate connections and structure tensors. We demonstrate that distinct conjugate connections such as $h,\widetilde{h},J$ and $J^{\ast }-$conjugates, may yield quasi-statistical manifolds under appropriate compatibility assumptions. In particular, we establish a number of geometric results under the assumptions of Codazzi coupling and $d^{\nabla }$-closedness. The novelty of our approach lies in combining the framework of metallic-like manifolds with quasi-statistical structures in the presence of torsion, thereby extending existing results in the literature and opening new directions for further research. Finally, we also present a theorem concerning the Tachibana operator, which highlights additional structural properties of the manifolds under consideration.

Keywords

References

Aktaş, B., Gezer, A., Durmaz, O., Investigating the interplay of Codazzi couplings and connections in metallic-like pseudo-Riemannian manifolds, Int. Electron. J. Geom., 18(2)(2025), 208–229.
Aktaş, B., Gezer, A., Durmaz, O., Quasi-statistical manifolds with almost Hermitian and almost anti-Hermitian structures, An. S¸ tiint¸. Univ. “Ovidius” Constant¸a Ser. Mat., 33(1)(2025), 5–32.
Amari, S., Nagaoka, H., Methods of Information Geometry, American Mathematical Society, Providence, Oxford University Press, Oxford, 2000.
Amari, S., Information geometry of the EM and em algorithms for neural networks, Neural Netw., 8(9)(1995), 1379–1408.
Amari, S., Differential-Geometrical Methods in Statistics, Lecture Notes in Statistics, Springer, 28, 1985.
Ay, N., Jost, J., Le, H.V., Schwachh¨ofer, L., Information Geometry, Springer, 2017.
Belkin, M., Niyogi, P., Sindhwani, V., Manifold regularization: A geometric framework for learning from labeled and unlabeled examples, J. Mach. Learn. Res., 7(85)(2006), 2399–2434.
Blaga, A.M., Nannicini, A., On the geometry of metallic pseudo-Riemannian structures, Riv. Math. Univ. Parma (N.S.), 11(1)(2020), 69–87.

Blaga, A.M., Nannicini, A., On curvature tensors of Norden and metallic pseudo-Riemannian manifolds, Complex Manifolds, 6(1)(2019), 150–159.
Caticha, A., The information geometry of space-time, Proceedings, 33(1)(2019), 3015.
Durmaz, O., Gezer, A., Conjugate connections and their applications on pure metallic metric geometries, Ricerche Mat., 74(2)(2025).
Durmaz, O., Gezer, A., Aktas¸, B., Unified framework for Hermitian-like and anti-Hermitian-like geometries (submitted)
Eguchi, S., A differential geometric approach to statistical inference on the basis of contrast functionals, Hiroshima Math. J., 15(2)(1985), 341–391.
Fei, T., Zhang, J., Interaction of Codazzi couplings with (para-) K¨ahler geometry, Results Math., 72(2)(2017), 2037–2056.
Gezer, A., Karaman, C., On metallic Riemannian structures, Turkish J. Math., 39(6)(2015), 954–962.
Hretcanu, C.E., Crasmareanu, M., Metallic structures on Riemannian manifolds, Rev. Un. Mat. Argentina, 54(2)(2013), 15–27.
Isham, C.J., Modern Differential Geometry for Physicists, World Scientific Publishing Co. Pte. Ltd., Singapore, 1999.
Kass, R.E., Vos, P.W., Geometrical Foundations of Asymptotic Inference, Wiley, 1997.
Kurose, T., On the divergence of 1-conformally flat statistical manifolds, Tohoku Math. J., 46(1994), 427–433.
Kurose, T., Statistical Manifolds Admitting Torsion, Geometry and Something, Fukuoka University, Fukuoka-shi, Japan, 2007.
Kullback, S., Leibler, R.A., On information and sufficiency, Ann. Math. Statist., 22(1)(1951), 79–86.
Manea, A., Metallic-like structures and metallic-like maps, Turkish J. Math., 47(5)(2023), 1539–1549.
Nomizu, K., Sasaki, T., Affine Differential Geometry: Geometry of Affine Immersions, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1994.
Nomizu, K., Simon, U., Notes on Conjugate Connections, in Geometry and Topology of Submanifolds IV,World Scientific Publishing Co. Inc. River Edge. NJ., 152-173, 1992.
Salimov, A., Tensor Operators and Their Applications, Nova Science Publishers, Inc., New York, 2013.
Schwenk-Schellschmidt, A., Simon, U., Codazzi-equivalent affine connections, Result Math., 56(1-4)(2009), 211–229.
Tachibana, S., Analytic tensor and its generalization, Tohoku Math. J., 12(2)(1960), 208–221.
Tricerri, F., Sulle varieta dotate di due strutture quasi complesse linearmente indipendenti, Riv. Mat. Univ. Parma (3), 3(1974), 349–358.
Vanzura, J., Integrability conditions for polynomial structures, Kodai Math. Sem. Rep., 27(1-2)(1976), 42–50.
Wakakuwa, H., On linearly independent almost complex structures in a differentiable manifold, Tohoku Math. J., 13(1961), 393–422.

Details

Primary Language

English

Subjects

Algebraic and Differential Geometry

Journal Section

Research Article

Authors

Aydın Gezer
0000-0001-7505-0385
Türkiye

Olgun Durmaz
0000-0002-0913-3307
Türkiye

Buşra Aktaş ^*
0000-0002-1285-7250
Türkiye

Publication Date

February 23, 2026

Submission Date

September 23, 2025

Acceptance Date

December 11, 2025

Published in Issue

Year 2026 Volume: 18 Number: 1

DOI

https://doi.org/10.47000/tjmcs.1789376

IZ

https://izlik.org/JA77LG87RB

Cite

RIS / Bibtex

APA

Gezer, A., Durmaz, O., & Aktaş, B. (2026). Extending Information Geometry: Quasi-Statistical Structures on Metallic-Like Pseudo-Riemannian Manifolds. Turkish Journal of Mathematics and Computer Science, 18(1), 248-266. https://doi.org/10.47000/tjmcs.1789376

AMA

1.Gezer A, Durmaz O, Aktaş B. Extending Information Geometry: Quasi-Statistical Structures on Metallic-Like Pseudo-Riemannian Manifolds. TJMCS. 2026;18(1):248-266. doi:10.47000/tjmcs.1789376

Chicago

Gezer, Aydın, Olgun Durmaz, and Buşra Aktaş. 2026. “Extending Information Geometry: Quasi-Statistical Structures on Metallic-Like Pseudo-Riemannian Manifolds”. Turkish Journal of Mathematics and Computer Science 18 (1): 248-66. https://doi.org/10.47000/tjmcs.1789376.

EndNote

Gezer A, Durmaz O, Aktaş B (February 1, 2026) Extending Information Geometry: Quasi-Statistical Structures on Metallic-Like Pseudo-Riemannian Manifolds. Turkish Journal of Mathematics and Computer Science 18 1 248–266.

IEEE

[1]A. Gezer, O. Durmaz, and B. Aktaş, “Extending Information Geometry: Quasi-Statistical Structures on Metallic-Like Pseudo-Riemannian Manifolds”, TJMCS, vol. 18, no. 1, pp. 248–266, Feb. 2026, doi: 10.47000/tjmcs.1789376.

ISNAD

Gezer, Aydın - Durmaz, Olgun - Aktaş, Buşra. “Extending Information Geometry: Quasi-Statistical Structures on Metallic-Like Pseudo-Riemannian Manifolds”. Turkish Journal of Mathematics and Computer Science 18/1 (February 1, 2026): 248-266. https://doi.org/10.47000/tjmcs.1789376.

JAMA

1.Gezer A, Durmaz O, Aktaş B. Extending Information Geometry: Quasi-Statistical Structures on Metallic-Like Pseudo-Riemannian Manifolds. TJMCS. 2026;18:248–266.

MLA

Gezer, Aydın, et al. “Extending Information Geometry: Quasi-Statistical Structures on Metallic-Like Pseudo-Riemannian Manifolds”. Turkish Journal of Mathematics and Computer Science, vol. 18, no. 1, Feb. 2026, pp. 248-66, doi:10.47000/tjmcs.1789376.

Vancouver

1.Aydın Gezer, Olgun Durmaz, Buşra Aktaş. Extending Information Geometry: Quasi-Statistical Structures on Metallic-Like Pseudo-Riemannian Manifolds. TJMCS. 2026 Feb. 1;18(1):248-66. doi:10.47000/tjmcs.1789376