Curvatures in contravariant warped product space

TL;DR

This paper introduces and computes sectional and qualar curvatures in contravariant warped product spaces, focusing on Poisson manifolds and dual space forms, with specific examples including hyperbolic and spherical geometries.

Contribution

It defines sectional and qualar curvatures in contravariant warped product spaces, extending geometric analysis to Poisson manifolds and dual space forms with explicit curvature calculations.

Findings

Sectional curvature formulas for contravariant warped products.

Introduction of null, spacelike, timelike 1-forms in dual space.

Explicit curvature computations for specific manifolds.

Abstract

In this article, we introduce the sectional curvature in contravariant warped product space $(M= M_{1}\times_{f_{1}}M_{2},\Pi,g^{f_{1}})$, where $\Pi=\Pi_1+\nu_{1}\Pi_2$). 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 $M$.