General Natural \((\alpha ,\varepsilon )\)-Structures
- 64 Downloads
We study in a unified way the \((\alpha ,\varepsilon )\)-structures of general natural lift type on the tangent bundle of a Riemannian manifold. We characterize the general natural \(\alpha \)-structures on the total space of the tangent bundle of a Riemannian manifold, and provide their integrability conditions (the base manifold is a space form and some involved coefficients are rational functions of the other ones). Then, we characterize the two classes (with respect to the sign of \(\alpha \varepsilon \)) of \((\alpha ,\varepsilon )\)-structures of general natural type on TM. The class \(\alpha \varepsilon =-1\) is characterized by some proportionality relations between the coefficients of the metric and those of the \(\alpha \)-structure, and in this case, the structure is almost Kählerian if and only if the first proportionality factor is the derivative of the second one. Moreover, the total space of the tangent bundle is a Kähler manifold if and only if it depends on three coefficients only (two coefficients of the integrable \(\alpha \)-structure and a proportionality factor).
KeywordsNatural lift \((\alpha , \varepsilon )\)-Structure Almost Hermitian metric Almost Kähler structure
Mathematics Subject ClassificationPrimary 53C15 53B35 53C55
The author wants to express her gratitude to Professor Fernando Etayo Gordejuela, for carefully reading the paper, and for his valuable suggestions, that led to the improvement of the paper.
- 4.Bejan, C.: A classification of the almost parahermitian manifolds. Proc. Conference on Diff. Geom. and Appl., Dubrovnik, 23–27 (1988)Google Scholar
- 5.Bejan, C.: Almost parahermitian structures on the tangent bundle of an almost paracohermitian manifold. Proc. Fifth Nat. Sem. Finsler and Lagrange spaces, Braşov, 105–109 (1988)Google Scholar
- 8.Cruceanu, V.: Selected Papers (37. Para-Hermitian and para-Kähler manifolds, pp. 339–387.), Editura PIM, Iaşi (2006)Google Scholar
- 22.Janyška, J.: Natural 2-forms on the tangent bundle of a Riemannian manifold. Rend. Circ. Mat. Palermo, Serie II, Supplemento, The Proceedings of the Winter School Geometry and Topology Srní-January 1992 32, 165–174 (1993)Google Scholar
- 23.Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I and II. Interscience, N. York (1963, 1969)Google Scholar
- 42.Teofilova, M.: Almost complex connections on almost complex manifolds with Norden metric. In: Sekigawa, K., Gerdjikov, V.S., Dimiev, S. (eds.) Trends in Differential Geometry, Complex Analysis and Mathematical Physics Proceedings of the 9th International Workshop on Complex Structures, Integrability and Vector Fields, Sofia, Bulgaria, 25–29 August 2008 pp. 231–240. World Scientific, Singapore (2009)Google Scholar