General Natural $$(\alpha ,\varepsilon )$$ ( α , ε ) -Structures

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 Kahlerian 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 Kahler manifold if and only if it depends on three coefficients only (two coefficients of the integrable $$\alpha $$ -structure and a proportionality factor).