Statistical structures on tangent bundles and tangent Lie groups

Yıl 2021, Cilt: 50 Sayı: 4, 1140 - 1154, 06.08.2021

Esmaeil Peyghan , Davood Seifipour Aydın Gezer

https://doi.org/10.15672/hujms.645070

Cited By: 4

Öz

Let $TM$ be a tangent bundle over a Riemannian manifold $M$ with a Riemannian metric $g$ and $TG$ be a tangent Lie group over a Lie group with a left-invariant metric $g$. The purpose of the paper is two folds. Firstly, we study statistical structures on the tangent bundle $TM$ equipped with two Riemannian $g$-natural metrics and lift connections. Secondly, we define a left-invariant complete lift connection on the tangent Lie group $TG$ equipped with metric $\tilde{g}$ introduced in [F. Asgari and H. R. Salimi Moghaddam, On the Riemannian geometry of tangent Lie groups, Rend. Circ. Mat. Palermo II. Series, 2018] and study statistical structures in this setting.

Anahtar Kelimeler

Metric, statistical structure, tangent bundle, tangent Lie group, lift connection

Kaynakça

• [1] M.T.K. Abbassi and M. Sarih, On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds, Diff. Geom. Appl. 22, 19–47, 2005.
• [2] S. Amari, Information geometry of the EM and em algorithms for neural networks, Neural Networks, 8 (9), 1379–1408, 1995.
• [3] F. Asgari and H.R. Salimi Moghaddam, On the Riemannian geometry of tangent Lie groups, Rend. Circ. Mat. Palermo, II. Ser. 67 (2), 185–195, 2018.
• [4] V. Balan, E. Peyghan and E. Sharahi, Statistical structures on the tangent bundle of a statistical manifold with Sasaki metric, Hacet. J. Math. Stat. 49 (1), 120–135, 2020.
• [5] M. Belkin, P. Niyogi and V. Sindhwani, Manifold regularization: a geometric framework for learning from labeled and unlabeled examples, J. Mach. Learn. Res. 7, 2399– 2434, 2006.
• [6] L. Bilen and A. Gezer, Some results on Riemannian g-natural metrics generated by classical lifts on the tangent bundle, Eurasian Math. J. 8 (4), 18–34, 2017.
• [7] T. Fei and J. Zhang, Interaction of Codazzi couplings with (Para-)Kähler geometry, Result Math. 72 (4), 2037–2056, 2017.
• [8] S. Gudmundsson and E. Kappos, On the geometry of the tangent bundles, Expo. Math. 20, 1–41, 2002.
• [9] S. Ianus, Statistical manifolds and tangent bundles, Sci. Bull. Univ. Politechnica of Bucharest Ser. D, 56, 29–34, 1994.
• [10] S.L. Lauritzen, Statistical manifolds, In: Differential Geometry in Statistical Inferences, IMS Lecture Notes Monogr. Ser. 10, Inst. Math. Statist. Hayward California, 96–163, 1987.
• [11] H. Matsuzoe and J.I. Inoguchi, Statistical structures on tangent bundles, APPS. Appl.Sci. 5 (1), 55–57, 2003.
• [12] K. Nomizu and T. Sasaki, Affine Differential Geometry: Geometry of Affine Immersions, vol. 111 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1994.
• [13] L. Nourmohammadifar, E. Peyghan and S. Uddin, Geometry of almost Kenmotsu Hom-Lie algebras, Quaest. Math. DOI: 10.2989/16073606.2021.1886194.
• [14] C.R. Rao, Information and accuracy attainable in the estimation of statistical parameters, Bull. Calcutta Math. Soc. 37, 81–91, 1945.
• [15] A. Schwenk-Schellschmidt and U. Simon, Codazzi-equivalent affine connections, Result Math. 56, 211–229, 2009.
• [16] K. Sun and S. Marchand-Maillet, An information geometry of statistical manifold learning, (Proceedings of the 31st International Conference on Machine Learning (ICML-14), 1–9, 2014.
• [17] K. Yano, and S. Ishihara, Tangent and cotangent bundles, Marcel Dekker, Inc., New York 1973.