Curvatures in contravariant warped product space

Abstract

In this article, we introduce the sectional curvature in contravariant warped product space $(M=M_1{\times }_{f_1}{M}_2,\hat{I},g^{f_1})$, where $\hat{I}={\hat{I}}_1+\hat{I}{½}_1{\hat{I}}_2$). After that we find the sectional curvature of $M$ for which $M_1$ and $M_2$ are Poisson manifolds of positive sectional curvatures. In dual space of $M$, we introduce the notion of null, spacelike, timelike $1-$ forms and then by using these forms, qualar curvature is defined. Finally, as an examples we obtain the sectional curvatures for $M_1=H_1^2$, $M_2=S_0^2,E_2^2$ and qualar curvature for $M$.