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Keywords:
Riemannian manifold; frame bundle; tangent bundle; natural metric
Riemannian manifold; frame bundle; tangent bundle; natural metric
Summary:
Let $(M,g)$ be a Riemannian manifold, $L(M)$ its frame bundle. We construct new examples of Riemannian metrics, which are obtained from Riemannian metrics on the tangent bundle $TM$. We compute the Levi–Civita connection and curvatures of these metrics.
References:
[1] Abbassi, M. T. K., Sarih, M.: On natural metrics on tangent bundles of Riemannian manifolds. Arch. Math. (Brno) 41 (1) (2005), 71–92. MR 2142144 | Zbl 1114.53015
[2] Baird, P., Wood, J. C.: Harmonic morphisms between Riemannian manifolds. Oxford University Press, Oxford, 2003. MR 2044031 | Zbl 1055.53049
[3] Benyounes, M., Loubeau, E., Wood, C. M.: Harmonic sections of Riemannian vector bundles, and metrics of Cheeger–Gromoll type. Differential Geom. Appl. 25 (3) (2007), 322–334. DOI 10.1016/j.difgeo.2006.11.010 | MR 2330461 | Zbl 1128.53037
[4] Benyounes, M., Loubeau, E., Wood, C. M.: The geometry of generalised Cheeger-Gromoll metrics. Tokyo J. Math. 32 (2) (2009), 287–312. DOI 10.3836/tjm/1264170234 | MR 2589947 | Zbl 1200.53025
[5] Cordero, L. A., de Leon, M.: Lifts of tensor fields to the frame bundle. Rend. Circ. Mat. Palermo (2) 32 (2) (1983), 236–271. DOI 10.1007/BF02844834 | MR 0729099
[6] Cordero, L. A., de Leon, M.: On the curvature of the induced Riemannian metric on the frame bundle of a Riemannian manifold. J. Math. Pures Appl. (9) 65 (1) (1986), 81–91. MR 0844241
[7] Dombrowski, P.: On the geometry of the tangent bundle. J. Reine Angew. Math. 210 (1962), 73–88. MR 0141050 | Zbl 0105.16002
[8] Kowalski, O., Sekizawa, M.: Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles. A classification. Bull. Tokyo Gakugei Univ. (4) 40 (1988), 1–29. MR 0974641
[9] Kowalski, O., Sekizawa, M.: On curvatures of linear frame bundles with naturally lifted metrics. Rend. Sem. Mat. Univ. Politec. Torino 63 (3) (2005), 283–295. MR 2202049 | Zbl 1141.53020
[10] Kowalski, O., Sekizawa, M.: On the geometry of orthonormal frame bundles. Math. Nachr. 281 (2008), no. 12, 1799–1809 281 (12) (2008), 1799–1809. DOI 10.1002/mana.200610715 | MR 2473330 | Zbl 1158.53015
[11] Kowalski, O., Sekizawa, M.: On the geometry of orthonormal frame bundles. II. Ann. Global Anal. Geom. 33 (4) (2008), 357–371. DOI 10.1007/s10455-007-9091-7 | MR 2395192 | Zbl 1141.53023
[12] Mok, K. P.: On the differential geometry of frame bundles of Riemannian manifolds. J. Reine Angew. Math. 302 (1978), 16–31. MR 0511689 | Zbl 0378.53016
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