Abstract
We construct a metrical framed \(f(3,-1)\)-structure on the (1, 1)-tensor bundle of a Riemannian manifold equipped with a Cheeger–Gromoll type metric and by restricting this structure to the (1, 1)-tensor sphere bundle, we obtain an almost metrical paracontact structure on the (1, 1)-tensor sphere bundle. Moreover, we show that the (1, 1)-tensor sphere bundles endowed with the induced metric are never space forms.
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Abbassi, M.T.K.; Kowalski, O.: On g-natural metrics with constant scalar curvature on unit tangent sphere bundles. In: Matsushita, Y. et al. (eds) Topics in Almost Hermitian Geometry and Related Fields, Proc in Honor of K. Sekigawa’s 60th birthday, pp 1–29. Hackensack, NJ: World Scientific (2005).
Abbassi, M.T.K.; Sarih, M.: On natural metrics on tangent bundles of Riemannian manifolds. Arch. Math. (Brno) 41, 71–92 (2005)
Cengiz, N.; Salimov, A.A.: Complete lifts of derivations to tensor bundles. Bol. Soc. Mat. Mexicana 8(3), 75–82 (2002)
Cheeger, J.; Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. 96(2), 413–443 (1972)
Druţă-Romaniuc, S.L.: Kaehler–Einstein structures of general natural lifted type on the cotangent bundles. Balkan J. Geom. Appl. 14(1), 30–39 (2009)
Druţă-Romaniuc, S.L.: General natural Riemannian almost product and para-Hermitian structures on tangent bundles. Taiwan. J. Math 16(2), 497–510 (2013)
Druţă-Romaniuc, S.L.: Riemannian almost product and para-hermitian cotangent bundles of general natural lift type. Acta Math. Hung. 139(3), 228–244 (2013)
Druţă-Romaniuc, S.L.; Oproiu, V.: Tangent sphere bundles of natural diagonal lift type. Balkan J. Geom. Appl. 15, 53–67 (2010)
Druţă-Romaniuc, S.L.; Oproiu, V.: Tangent sphere bundles which are \(\eta \)-Einstein. Balkan J. Geom. Appl. 16(2), 48–61 (2011)
Druţă-Romaniuc, S.L.; Oproiu, V.: The holomorphic \(\phi \)-sectional curvature of tangent sphere bundles with Sasakian structures. An. ştiinţ. Univ. Al. I. Cuza Iaşi, Mat. 57(Suppl.), 75–86 (2011)
Kaneyuki, S.; Kozai, M.: Paracomplex structures and affine symmetric spaces. Tokyo J. Math. 8, 81–98 (1985)
Kolar, I.; Michor, P.W.; Slovak, J.: Natural operations in differential geometry. Springer, Berlin (1993)
Kowalski, O.; Sekizawa, M.: Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles—a classification. Bull. Tokyo Gakugei Univ. 40(4), 1–29 (1988)
Ledger, A.J.; Yano, K.: Almost complex structures on the tensor bundles. J. Diff. Geom. 1, 355–368 (1967)
Munteanu, M.I.: New CR-structures on the unit tangent bundle. An. Univ. Timisoara Ser. Mat. Inform. 38(no. 1), 99–110 (2000)
Munteanu, M.I.: Some aspects on the geometry of the tangent bundles and tangent sphere bundles of a Riemannian manifold. Mediterr. J. Math. 5(1), 43–59 (2008)
Peyghan, E.; Tayebi, A.; Nourmohammadi Far, L.: Cheeger–Gromoll type metric on (1, 1)-tensor bundle. J. Contemp. Math. Anal. 6, 59–70 (2013)
Salimov, A.A.; Cengiz, N.: Lifting of Riemannian metrics to tensor bundles. Russ. Math. (IZ. VUZ.) 47(11), 47–55 (2003)
Salimov, A.; Gezer, A.: On the geometry of the \((1, 1)\)-tensor bundle with Sasaki type metric. Chin. Ann. Math. 32(B3), 1–18 (2011)
Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds. Tohoku Math. J. 10, 338–358 (1958)
Sekizawa, M.: Curvatures of tangent bundles with Cheeger–Gromoll metric. Tokyo J. Math. 14(2), 407–417 (1991)
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Peyghan, E., Nourmohammadifar, L. & Tayebi, A. (1,1)-Tensor sphere bundle of Cheeger–Gromoll type. Arab. J. Math. 6, 315–327 (2017). https://doi.org/10.1007/s40065-017-0172-6
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DOI: https://doi.org/10.1007/s40065-017-0172-6