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General Natural \((\alpha ,\varepsilon )\)-Structures

  • S. L. Druţă-Romaniuc
Article
  • 35 Downloads

Abstract

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).

Keywords

Natural lift \((\alpha , \varepsilon )\)-Structure Almost Hermitian metric Almost Kähler structure 

Mathematics Subject Classification

Primary 53C15 53B35 53C55 

Notes

Acknowledgements

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.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsTechnical University “Gheorghe Asachi” of IaşiIasiRomania

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