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.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle.
📊
Key Findings & Data:Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle.
💡
Conclusion & Impact: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.
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.
Altunbaş, M. (2023). Statistical structures and Killing vector fields on tangent bundles with respect to two different metrics. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Volume: 72 Issue: 3. https://doi.org/10.31801/cfsuasmas.1160135
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
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
Yano, K., Ishihara, S., Tangent and cotangent bundles, Marcel Dekker Inc., New York, 1973.
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