Statistical structures and Killing vector fields on tangent bundles with respect to two different metrics

Year 2023, Volume: 72 Issue: 3, 815 - 825, 30.09.2023

Murat Altunbaş

https://doi.org/10.31801/cfsuasmas.1160135

Abstract

Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. The purpose of this paper is to study statistical structures on $TM$ with respect to the metrics $G_{1}=^{c}g+^{v}(fg)$ and $G_{2}=^{s}g_{f}+^{h}g,\ $ where $f$ is a smooth function on $M,$ $^{c}g$ is the complete lift of $g$, $^{v}(fg)$ is the vertical lift of $fg$, $^{s}g_{f}$ is a metric obtained by rescaling the Sasaki metric by a smooth function $f$ and $^{h}g$ is the horizontal lift of $g.$ Moreover, we give some results about Killing vector fields on $TM$ with respect to these metrics.

Keywords

Statistical manifold, Riemannian metric, tangent bundle

References

• Abbassi, M. T. K., Sarih, M., On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. (Brno), 41 (2005), 71-92.
• Altunbaş, M, Gezer, A., Bilen, L., Remarks about the Kaluza-Klein metric on tangent bundle, Int. J. Geom. Met. Mod. Phys., 16(3) (2019), 1950040. https://doi.org/10.1142/S0219887819500403
• Amari, S., Differential geometric methods in statistics- Lect. Notes in Stats., Springer, New York, 1985.
• Anastasiei, M., Locally conformal Kaehler structures on tangent bundle of a space form, Libertas Math., 19 (1999), 71-76.
• Balan, V., Peyghan, E., Sharahi, E., Statistical structures on the tangent bundle of a statistical manifold with Sasaki metric, Hacettepe J. Math. Stat., 49(1) (2020), 120-135. https://doi.org/10.15672/HJMS.2019.667
• Dombrowski, P., On the geometry of the tangent bundle, J. Reine Angew. Math., 210 (1962), 73-88. https://doi.org/10.1515/crll.1962.210.73
• Gezer, A., Altunbaş, M., Some notes concerning Riemannian metrics of Cheeger Gromoll type, J. Math. Anal. App., 396(1) (2012), 119-132. https://doi.org/10.1016/j.jmaa.2012.06.011
• Gezer, A., Bilen, L., Karaman, C¸ ., Altunba¸s, M., Curvature properties of Riemannian metrics of the forms Sgf +Hg on the tangent bundle over a Riemannian manifold (M,g), Int. Elec. J. Geo., 8(2) (2015), 181-194. https://doi.org/10.36890/iejg.592306
• Gezer, A., Ozkan, M., Notes on the tangent bundle with deformed complete lift metric, Turkish J. Math., 38 (2014), 1038-1049. https://doi.org/10.3906/mat-1402-30
• Lauritzen, S., Statistical manifolds. In Differential geometry in statistical inference, IMS lecture notes monograph series (10), Institute of Mathematical Statistics, Hyward, CA, USA, 96-163, 1987.
• Peyghan, E., Seifipour, D., Blaga, A., On the geometry of lift metrics and lift connections on the tangent bundle, Turkish J. Math., 46(6) (2022), 2335-2352. https://doi.org/10.55730/1300-0098.3272
• Peyghan, E., Seifipour, D., Gezer, A., Statistical structures on tangent bundles and Lie groups. Hacettepe J. Math. Stat., 50 (2021), 1140-1154. https://doi.org/10.15672/hujms.645070
• Salimov, A., Kazimova, S., Geodesics of the Cheeger-Gromoll metric, Turkish J. Math., 33(1) (2009), 99-105. https://doi.org/10.3906/mat-0804-24
• Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J., 10 (1958), 338-358. https://doi.org/10.2748/tmj/1178244668
• Sekizawa, M., Curvatures of tangent bundles with Cheeger-Gromoll metric, Tokyo J. Math. 14(2) (1991), 407-417. https://doi.org/10.3836/tjm/1270130381
• Yano, K., Ishihara, S., Tangent and cotangent bundles, Marcel Dekker Inc., New York, 1973.