GEOMETRY AND CURVATURE ANALYSIS OF THE BERGER-TYPE CHEEGER-GROMOLL METRIC ON TANGENT BUNDLES OVER ANTI-PARAKÄHLER MANIFOLDS

Abderrahim Zagane, Aydin Gezer, Saadia Chaoui

DOI Number
https://doi.org/10.22190/FUMI231128007Z
First page
069
Last page
094

Abstract


In this paper, we introduce a novel metric known as a Berger-type Cheeger-Gromoll metric on the tangent bundle TM over an anti-paraKähler manifold $(M,\varphi ,g)$. In this paper, we introduce a novel metric known as a Berger-type Cheeger-Gromoll metric on the tangent bundle TM over an anti-paraKähler manifold  $\varphi $-unit tangent bundle, which is endowed with the Berger-type Cheeger-Gromoll metric. Within this context, we provide the Levi-Civita connection and detail all forms of the Riemannian curvature tensors associated with this metric.


Keywords

Tangent bundle, anti-paraKähler manifold, Riemannian curvature tensor, φ-unit tangent bundle, Berger-type Cheeger-Gromoll metric

Full Text:

PDF

References


M. T. K. Abbassi and M. Sarih: On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. 41 (2005), 71 –92.

M. Altunbas, L. Bilen and A. Gezer: Remarks about the Kaluza-Klein metric on tangent bundle, Int. J. Geom. Methods Mod. Phys. 16 (3) (2019), 1950040, 13 pp.

M. Altunbas, R. Simsek and A. Gezer: A study concerning Berger-type deformed Sasaki metric on the tangent bundle, Zh. Mat. Fiz. Anal.Geom. 15 (4) (2019), 435–447.

N. Boussekkine and A. Zagane: On deformed-sasaki metric and harmonicity in tangent bundles, Commun. Korean Math. Soc. 35 (3) (2020), 1019-1035.

J. Cheeger and D. Gromoll: On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96 (1972), 413-443.

P. Dombrowski: On the geometry of the tangent bundle, J. Reine Angew. Math. 210 (1962), 73–88.

A. Gezer and M. Ozkan: Notes on the tangent bundle with deformed complete lift metric, Turkish J. Math. 38 (2014), 1038-1049.

A. Gezer: On the tangent bundle with deformed Sasaki metric, Int. Electron. J. Geom. 6 (2) (2013), 19-31.

A. Gezer and M. Altunbas: Notes on the rescaled Sasaki type metric on the cotangent bundle, Acta Math. Sci. Ser. B (Engl. Ed.) 34 (1) (2014), 162-174.

A. Gezer and M. Altunbas: On the geometry of the rescaled Riemannian metric on tensor bundles of arbitrary type, Kodai Math. J. 38 (1) (2015), 37-64.

S. Gudmundsson and E. Kappos: On the geometry of the tangent bundle with the Cheeger-Gromoll metric, Tokyo J. Math. 25 (1) (2002), 75–83.

O. Kowalski: Curvature of the induced Riemannian metric of the tangent bundle of Riemannian manifold, J. Reine Angew.Math. 250 (1971), 124-129.

F. Latti, M. Djaa and A. Zagane: Mus-Sasaki metric and harmonicity, Math. Sci. Appl. E-Notes 6 (1) (2018), 29–36.

E. Musso and F. Tricerri: Riemannian metrics on tangent bundles, Ann. Mat. Pura. Appl. 150 (4) (1988), 1–19.

A. A. Salimov, A. Gezer and K. Akbulut: Geodesics of Sasakian metrics on tensor bundles, Mediterr. J. Math. 6 (2) (2009), 135–147.

A. A. Salimov, M. Iscan and F. Etayo: Para-holomorphic B-manifold and its properties, Topology Appl. 154 (4) (2007), 925–933.

S. Sasaki: On the differential geometry of tangent bundles of Riemannian manifolds II, Tohoku Math. J. 14 (2) (1962), 146–155.

M. Sekizawa: Curvatures of tangent bundles with Cheeger-Gromoll metric, Tokyo J. Math. 14 (2) (1991), 407–417.

A. Yampolsky: On geodesics of tangent bundle with fiberwise deformed Sasaki metric over Kahlerian manifolds, Zh. Mat. Fiz. Anal. Geom. 8 (2) (2012), 177–189.

K. Yano and S. Ishihara: Tangent and cotangent bundles, M. Dekker, New York, 1973.

A. Zagane and A. Gezer: Vertical Rescaled Cheeger-Gromoll metric and harmonicity on the cotangent bundle, Adv. Stud. Euro-Tbil. Math. J. 15 (3) (2022), 11-29.

A. Zagane: Berger-type deformed Sasaki metric and harmonicity on the cotangent bundle, Int. Electron. J. Geom. 14 (1) (2021), 183-195.

A. Zagane and N. Boussekkine: Some almost paracomplex structures on the tangent bundle with vertical rescaled Berger deformation metric, Balkan J. Geom. Appl. 26 (1) (2021), 124–140.

A. Zagane: Berger-type deformed Sasaki metric on the cotangent bundle, Commun. Korean Math. Soc. 36 (3) (2021), 575–592.

A. Zagane: Vertical rescaled Berger deformation metric on the tangent bundle, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 41 (4) (2021), 166-180.




DOI: https://doi.org/10.22190/FUMI231128007Z

Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)