A Short Note on a Mus-Cheeger-Gromoll Type Metric
Abstract
In this paper, we first show that the complete lift $U^{c}$ to $TM$ of a vector field $U$ on $M$ is an infinitesimal fiber-preserving conformal transformation if and only if $U$ is an infinitesimal homothetic transformation of $(M,g)$. Here, $(M, g)$ is a Riemannian manifold and $TM$ is its tangent bundle with a Mus-Cheeger-Gromoll type metric $\tilde{g}$. Secondly, we search for some conditions under which $\left(\overset{h}{\nabla},\tilde{g}\right)$ is a Codazzi pair on $TM$ when $(\nabla, g)$ is a Codazzi pair on $M$ where $\overset{h}{\nabla}$ is the horizontal lift of a linear connection $\nabla$ on $M$. We finally discuss the need for further research.
Keywords
Codazzi pair, infinitesimal fiber-preserving conformal transformation, infinitesimal homothetic transformation, Mus-Cheeger-Gromoll type metric, tangent bundle
References
- S. Sasaki, On the Differential Geometry of Tangent Bundles of Riemannian Manifolds, Tohoku Mathematical Journal 10 (1958) 338–358.
- E. Musso, F. Tricerri, Riemannian Metrics on Tangent Bundles, Annali di Matematica Pura ed Applicata 150 (4) (1988) 1–19.
- B. V. Zayatuev, On Geometry of Tangent Hermitian Surface, Webs and Quasigroups T. S. U. (1995) 139–143.
- L. Belarbi, H. El Hendi, Geometry of Twisted Sasaki Metric, Journal of Geometry and Symmetry in Physics 53 (2019) 1–19.
- A. Zagane, M. Djaa, Geometry of Mus-Sasaki Metric, Communications in Mathematics 26 (2) (2018) 113–126.
- J. Wang, Y. Wang, On the Geometry of Tangent Bundles with the Rescaled Metric (2011), https://arxiv.org/abs/1104.5584.
- M. Benyounes, M., E. Loubeau, C. M. Wood, The Geometry of Generalised Cheeger-Gromoll Metrics, Tokyo Journal of Mathematics 32 (2009) 1–26.
- A. Gezer, M. Altunba¸s, Some Notes Concerning Riemannian Metrics of Cheeger Gromoll Type, Journal of Mathematical Analysis and Applications 396 (1) (2012) 119–132.
- F. Latti, M. Djaa, On the Geometry of the Mus-Cheeger-Gromoll Metric, Bulletin Transilvania University Brasov Series III 64 (2) (2022) 121–138.
- A. Gezer, L. Bilen, On Infinitesimal Conformal Transformations of the Tangent Bundles with Respect to the Cheeger-Gromoll Metric, Analele Stiintifice ale Universitatii Ovidius Constanta 20 (1) (2012) 113–128.