Research Article
BibTex RIS Cite

Conformal Geometric Solitons on the Tangent Bundle of a Statistical Manifold

Year 2025, Volume: 18 Issue: 1, 86 - 96, 24.04.2025

Lokman Bilen , Aydın Gezer

Abstract

Let $(M_{n},\nabla ,g)$ denote a statistical manifold equipped with a torsion-free linear connection $\nabla $ and a (pseudo-) Riemannian metric $%g $. The tangent bundle $TM$ of the statistical manifold $(M_{n},\nabla ,g)$ is endowed with a twisted Sasaki metric, denoted as $G$. The objective of this paper is to explore conformal Ricci, conformal Yamabe, and conformal Ricci-Yamabe solitons on the tangent bundle $TM$ concerning the twisted Sasaki metric $G$.

Keywords

Conformal soliton tangent bundle twisted Sasaki metric statistical manifold

Thanks

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

References

  • Abbassi, M. T. K. and Sarih, M.: On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds. Difer. Geom. Appl., 22 (1), 19-47 (2005).
  • Altunbaş M.: Ricci-Yamabe solitons on the Lie group H2R. International Journal of Maps in Mathematics, 7 (2), 217-223 (2024).
  • Altunbaş, M.: Conformal Yamabe Solitons on Tangent Bundles with Complete Lifts of Some Special Connections. Proceedings of the Bulgarian Academy of Sciences, 76 (8), 1176–1186 (2023).
  • Amari, S.: Information geometry of the EM and em algorithms for neural networks. Neural Networks, 8 (9), 1379-1408 (1995).
  • Amari, S. and Nagaoka, H.: Methods of information geometry. American Mathematical Society, (2000).
  • Amari, S.: Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics, 28, Springer, New York, (1985).
  • Barbosa, E. and Ribeiro Jr., E.: On conformal solutions of the Yamabe flow. Arch. Math. 101, 79-89 (2013).
  • Basu, N., Bhattacharyya, A.: Conformal Ricci soliton in Kenmotsu manifold. Global Journal of Advanced Research on Classical and Modern Geometries, 4 (1), 15-21 (2015).
  • Belkin, M., Niyogi, P. and Sindhwani, V.: Manifold regularization: a geometric framework for learning from labeled and unlabeled examples. Journal of Machine Learning Research, 7, 2399-2434 (2006).
  • Bilen, L. and Gezer, A. Some results on Riemannian g-natural metrics generated by classical lifts on the tangent bundle. Eurasian Math. J. 8 (4), 18–34 (2017).
  • Blaga, A. M. and Perkta¸s, S. Y. Remarks on almost n-Ricci solitons in (ϵ)-para Sasakian manifolds. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68 (2), 1621-1628 (2019).
  • Caticha, A.: Geometry from information geometry. AIP Conf. Proc. 1757, 030001 (2016).
  • Caticha, A.: The information geometry of space and time. AIP Conf. Proc. 803, 355-365 (2005).
  • Catino, G. and Mazzieri, L.: Gradient Einstein solitons. Nonlinear Anal., 132, 66-94 (2016).
  • Fei, T. and Zhang, J.: Interaction of Codazzi couplings with (Para-)Kähler geometry. Result Math. 72 (4), 2037-2056 (2017).
  • Fischer, A. E.: An introduction to conformal Ricci flow. Class. Quantum Grav. 21 (3), 171-218 (2004).
  • Gezer, A., Bilen L. and De U. C.: Conformal vector fields and geometric solitons on the tangent bundle with the ciconia metric. Filomat., 37 (24), 8193-8204 (2023).
  • Güler, S. and Crasmareanu, M.: Ricci-Yamabe maps for Riemannian flows and their volume variation and volume entropy. Turkish J. Math. 43 (5), 2631–2641 (2019).
  • Hamilton, R. S.: The Ricci flow on surfaces. Contemp. Math. 71, 237-262 (1988).
  • Hamilton, R. S.: Three manifold with positive Ricci curvature. J. Differential Geom. 17 (2), 255-306 (1982).
  • Lauritzen, S. L.: Statistical manifolds In: Differential Geometry in Statistical Inferences. IMS Lecture Notes Monogr. Ser. 10, Inst. Math. Statist. Hayward California, 96-163 (1987).
  • Li, Y., Bilen, L. and Gezer, A.: Analyzing Curvature Properties and Geometric Solitons of the Twisted Sasaki Metric on the Tangent Bundle over a Statistical Manifold. Mathematics. 12, 1395 (2024). https://doi.org/10.3390/math12091395
  • Li, Y., Gezer, A., Karakas, E.: Exploring Conformal Soliton Structures in Tangent Bundles with Ricci-Quarter Symmetric Metric Connections. Mathematics. 12, 2101 (2024). https://doi.org/10.3390/math12132101
  • Matsuzoe, H.: Statistical manifolds and affine differential geometry. Advanced Studies in Pure Mathematics 57, Probabilistic Approach to Geometry, pp. 303-321 (2010).
  • Rao, C. R.: Information and Accuracy Attainable in the Estimation of Statistical Parameters. Bull. Calcutta Math. Soc. 37, 81-91 (1945).
  • Roy, S., Dey, S. and Bhattacharyya, A.: Conformal Yamabe soliton and *-Yamabe soliton with torse forming potential vector field. Mat. Vesnik 73 (4), 282-292 (2021).
  • Roy, S., Bhattacharyya, A.: A Kenmotsu Metric as a*-conformal Yamabe Soliton with Torse Forming Potential Vector Field. Acta Math. Sci. 37, 1896-1908 (2021).
  • Schwenk-Schellschmidt, A. and Simon, U.: Codazzi-equivalent affine connections. Result Math. 56, 211-229 (2009).
  • Sun, K. and Marchand-Maillet, S.: An information geometry of statistical manifold learning. Proceedings of the 31st International Conference on Machine Learning (ICML-14), 1-9 (2014).
  • Yamauchi, K.: On infinitesimal conformal transformations of the tangent bundles over Riemannian manifolds. Ann Rep. Asahikawa. Med. Coll., 15, 1-10 (1994).
  • Yano, K. and Ishihara, S.: Tangent and Cotangent Bundles. Marcel Dekker, Inc, New York, USA, (1973).

Year 2025, Volume: 18 Issue: 1, 86 - 96, 24.04.2025

Lokman Bilen , Aydın Gezer

Abstract

References

  • Abbassi, M. T. K. and Sarih, M.: On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds. Difer. Geom. Appl., 22 (1), 19-47 (2005).
  • Altunbaş M.: Ricci-Yamabe solitons on the Lie group H2R. International Journal of Maps in Mathematics, 7 (2), 217-223 (2024).
  • Altunbaş, M.: Conformal Yamabe Solitons on Tangent Bundles with Complete Lifts of Some Special Connections. Proceedings of the Bulgarian Academy of Sciences, 76 (8), 1176–1186 (2023).
  • Amari, S.: Information geometry of the EM and em algorithms for neural networks. Neural Networks, 8 (9), 1379-1408 (1995).
  • Amari, S. and Nagaoka, H.: Methods of information geometry. American Mathematical Society, (2000).
  • Amari, S.: Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics, 28, Springer, New York, (1985).
  • Barbosa, E. and Ribeiro Jr., E.: On conformal solutions of the Yamabe flow. Arch. Math. 101, 79-89 (2013).
  • Basu, N., Bhattacharyya, A.: Conformal Ricci soliton in Kenmotsu manifold. Global Journal of Advanced Research on Classical and Modern Geometries, 4 (1), 15-21 (2015).
  • Belkin, M., Niyogi, P. and Sindhwani, V.: Manifold regularization: a geometric framework for learning from labeled and unlabeled examples. Journal of Machine Learning Research, 7, 2399-2434 (2006).
  • Bilen, L. and Gezer, A. Some results on Riemannian g-natural metrics generated by classical lifts on the tangent bundle. Eurasian Math. J. 8 (4), 18–34 (2017).
  • Blaga, A. M. and Perkta¸s, S. Y. Remarks on almost n-Ricci solitons in (ϵ)-para Sasakian manifolds. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68 (2), 1621-1628 (2019).
  • Caticha, A.: Geometry from information geometry. AIP Conf. Proc. 1757, 030001 (2016).
  • Caticha, A.: The information geometry of space and time. AIP Conf. Proc. 803, 355-365 (2005).
  • Catino, G. and Mazzieri, L.: Gradient Einstein solitons. Nonlinear Anal., 132, 66-94 (2016).
  • Fei, T. and Zhang, J.: Interaction of Codazzi couplings with (Para-)Kähler geometry. Result Math. 72 (4), 2037-2056 (2017).
  • Fischer, A. E.: An introduction to conformal Ricci flow. Class. Quantum Grav. 21 (3), 171-218 (2004).
  • Gezer, A., Bilen L. and De U. C.: Conformal vector fields and geometric solitons on the tangent bundle with the ciconia metric. Filomat., 37 (24), 8193-8204 (2023).
  • Güler, S. and Crasmareanu, M.: Ricci-Yamabe maps for Riemannian flows and their volume variation and volume entropy. Turkish J. Math. 43 (5), 2631–2641 (2019).
  • Hamilton, R. S.: The Ricci flow on surfaces. Contemp. Math. 71, 237-262 (1988).
  • Hamilton, R. S.: Three manifold with positive Ricci curvature. J. Differential Geom. 17 (2), 255-306 (1982).
  • Lauritzen, S. L.: Statistical manifolds In: Differential Geometry in Statistical Inferences. IMS Lecture Notes Monogr. Ser. 10, Inst. Math. Statist. Hayward California, 96-163 (1987).
  • Li, Y., Bilen, L. and Gezer, A.: Analyzing Curvature Properties and Geometric Solitons of the Twisted Sasaki Metric on the Tangent Bundle over a Statistical Manifold. Mathematics. 12, 1395 (2024). https://doi.org/10.3390/math12091395
  • Li, Y., Gezer, A., Karakas, E.: Exploring Conformal Soliton Structures in Tangent Bundles with Ricci-Quarter Symmetric Metric Connections. Mathematics. 12, 2101 (2024). https://doi.org/10.3390/math12132101
  • Matsuzoe, H.: Statistical manifolds and affine differential geometry. Advanced Studies in Pure Mathematics 57, Probabilistic Approach to Geometry, pp. 303-321 (2010).
  • Rao, C. R.: Information and Accuracy Attainable in the Estimation of Statistical Parameters. Bull. Calcutta Math. Soc. 37, 81-91 (1945).
  • Roy, S., Dey, S. and Bhattacharyya, A.: Conformal Yamabe soliton and *-Yamabe soliton with torse forming potential vector field. Mat. Vesnik 73 (4), 282-292 (2021).
  • Roy, S., Bhattacharyya, A.: A Kenmotsu Metric as a*-conformal Yamabe Soliton with Torse Forming Potential Vector Field. Acta Math. Sci. 37, 1896-1908 (2021).
  • Schwenk-Schellschmidt, A. and Simon, U.: Codazzi-equivalent affine connections. Result Math. 56, 211-229 (2009).
  • Sun, K. and Marchand-Maillet, S.: An information geometry of statistical manifold learning. Proceedings of the 31st International Conference on Machine Learning (ICML-14), 1-9 (2014).
  • Yamauchi, K.: On infinitesimal conformal transformations of the tangent bundles over Riemannian manifolds. Ann Rep. Asahikawa. Med. Coll., 15, 1-10 (1994).
  • Yano, K. and Ishihara, S.: Tangent and Cotangent Bundles. Marcel Dekker, Inc, New York, USA, (1973).
There are 31 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Lokman Bilen 0000-0001-8240-5359

Aydın Gezer 0000-0001-7505-0385

Early Pub Date April 20, 2025
Publication Date April 24, 2025
Submission Date June 11, 2024
Acceptance Date October 8, 2024
Published in Issue Year 2025 Volume: 18 Issue: 1

Cite

APA Bilen, L., & Gezer, A. (2025). Conformal Geometric Solitons on the Tangent Bundle of a Statistical Manifold. International Electronic Journal of Geometry, 18(1), 86-96.
AMA Bilen L, Gezer A. Conformal Geometric Solitons on the Tangent Bundle of a Statistical Manifold. Int. Electron. J. Geom. April 2025;18(1):86-96.
Chicago Bilen, Lokman, and Aydın Gezer. “Conformal Geometric Solitons on the Tangent Bundle of a Statistical Manifold”. International Electronic Journal of Geometry 18, no. 1 (April 2025): 86-96.
EndNote Bilen L, Gezer A (April 1, 2025) Conformal Geometric Solitons on the Tangent Bundle of a Statistical Manifold. International Electronic Journal of Geometry 18 1 86–96.
IEEE L. Bilen and A. Gezer, “Conformal Geometric Solitons on the Tangent Bundle of a Statistical Manifold”, Int. Electron. J. Geom., vol. 18, no. 1, pp. 86–96, 2025.
ISNAD Bilen, Lokman - Gezer, Aydın. “Conformal Geometric Solitons on the Tangent Bundle of a Statistical Manifold”. International Electronic Journal of Geometry 18/1 (April 2025), 86-96.
JAMA Bilen L, Gezer A. Conformal Geometric Solitons on the Tangent Bundle of a Statistical Manifold. Int. Electron. J. Geom. 2025;18:86–96.
MLA Bilen, Lokman and Aydın Gezer. “Conformal Geometric Solitons on the Tangent Bundle of a Statistical Manifold”. International Electronic Journal of Geometry, vol. 18, no. 1, 2025, pp. 86-96.
Vancouver Bilen L, Gezer A. Conformal Geometric Solitons on the Tangent Bundle of a Statistical Manifold. Int. Electron. J. Geom. 2025;18(1):86-9.
 Download Cover Image