Abstract
In the present article we consider a Lie group G equipped with a left invariant Riemannian metric g. Then, by using complete and vertical lifts of left invariant vector fields we induce a left invariant Riemannian metric \(\widetilde{g}\) on the tangent Lie group TG. The Levi-Civita connection and sectional curvature of \((TG,\widetilde{g})\) are given, in terms of Levi-Civita connection and sectional curvature of (G, g). Then, we present Levi-Civita connection, sectional curvature and Ricci tensor formulas of \((TG,\widetilde{g})\) in terms of structure constants of the Lie algebra of G. Finally, some examples of tangent Lie groups of strictly negative and non-negative Ricci curvatures are given.
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References
Crampin, M., Cantrijn, F., Sarlet, W.: Lifting geometric objects to a cotangent bundle, and the geometry of the cotangent bundle of a tangent bundle. J. Geom. Phys. 4(4), 469–492 (1987)
Eberlein, P.: Left invariant geometry of Lie groups. Cubo 6(1), 427–510 (2004)
Gudmundsson, S., Kappos, E.: On the geometry of tangent bundles. Expo. Math. 20, 1–41 (2002)
Ha, K.Y., Lee, J.B.: Left invariant metrics and curvatures on simply connected three-dimensional Lie groups. Math. Nachr. 282(6), 868–898 (2009)
Hilgert, J., Neeb, K.-H.: Structure and Geometry of Lie Groups, Springer Monographs in Mathematics (2012)
Hindeleh, F.Y.: Tangent and Cotangent Bundles, Automorphism Groups and Representations of Lie Groups, Electronic thesis or dissertation. University of Toledo (2006). https://etd.ohiolink.edu/
Kowalski, O., Sekizawa, M.: Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles. Bull. Tokyo Gakugei Univ. Sect. IV 40, 1–29 (1988)
Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)
Morandi, G., Ferrario, C., Lo Vecchio, G., Marmo, G., Rubano, C.: The inverse problem in the calculus of variations and the geometry of the tangent bundle. Phys. Rep. 188, 147284 (1990)
Ocak, F., Salimov, A.A.: Geometry of the cotangent bundle with Sasakian metrics and its applications. Proc. Indian Acad. Sci. (Math. Sci.) 124(3), 427–436 (2014)
O’Neill, B.: Semi-Riemannian Geometry, with Applications to Relativity. Academic Press, New York (1983)
Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds. Tohoku Math. J. 10, 338–354 (1958)
Sekizawa, M.: On complete lifts of reductive homogeneous spaces and generalized symmetric spaces. Czechoslovak Math. J. 36(4), 516–534 (1986)
Yano, K., Ishihara, S.: Tangent and Cotangent Bundles, Volume 16 of Pure and Applied Mathematics. Marcel Dekker, Inc., New York City (1973)
Yano, K., Kobayashi, S.: Prolongations of tensor fields and connections to tangent bundles I. J. Math. Soc. Jpn. 18(2), 194–210 (1966)
Yano, K., Kobayashi, S.: Prolongations of tensor fields and connections to tangent bundles II. J. Math. Soc. Jpn. 18(3), 236–246 (1966)
Yano, K., Kobayashi, S.: Prolongations of tensor fields and connections to tangent bundles III. J. Math. Soc. Jpn. 19(4), 486–488 (1967)
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We are grateful to the office of Graduate Studies of the University of Isfahan for their support.
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Asgari, F., Salimi Moghaddam, H.R. On the Riemannian geometry of tangent Lie groups. Rend. Circ. Mat. Palermo, II. Ser 67, 185–195 (2018). https://doi.org/10.1007/s12215-017-0304-z
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DOI: https://doi.org/10.1007/s12215-017-0304-z
Keywords
- Left invariant Riemannian metric
- Tangent Lie group
- Complete and vertical lifts
- Sectional and Ricci curvatures