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Old and new structures on the tangent bundle. (English) Zbl 1126.53020
Mladenov, Ivaïlo (ed.) et al., Proceedings of the 8th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 9–14, 2006. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-8495-37-0/pbk). 264-278 (2007).
The paper under review extends definitions and results on Sasakian and Cheeger-Gromoll metrics. More precisely, the author constructs the metric introduced by M. Anastasiei [Libertas Math. 19, 71–76 (1999; Zbl 0982.53064)], using the general method of Riemannian metrics on the tangent bundle TM given by E. Musso and F. Tricerri [Ann. Mat. Pura Appl. (4) 150, 1–19 (1988; Zbl 0658.53045)]. Furthermore, with the help of a compatible almost complex structure he obtains conditions under which TM is Kählerian, almost Kählerian, or locally conformal Kählerian. The paper ends with conditions when TM is of constant scalar or sectional curvature and with examples of such metrics.
53C15Differential geometric structures on manifolds
53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55Hermitian and Kählerian manifolds (global differential geometry)