# Some Flatness Conditions on Normal Metric Contact Pairs

## TL;DR

This paper investigates the geometric properties of normal metric contact pair manifolds under various flatness conditions of curvature tensors, revealing their Einstein nature and specific curvature characteristics.

## Contribution

It establishes new results linking flatness of conformal, concircular, and quasi-conformal curvature tensors to Einstein manifolds with specific scalar curvatures in the context of normal metric contact pairs.

## Key findings

- Conformal flatness implies Einstein manifold with negative scalar curvature and positive sectional curvature.
- Concircular flatness implies Einstein manifold.
- Quasi-conformal flatness implies Einstein manifold with positive scalar curvature and constant curvature.

## Abstract

In this paper the geometry of normal metric contact pair manifolds is studied under the flatness of conformal, concircular and quasi-conformal curvature tensors. It is proved that a conformal flat normal metric contact pair manifold is an Einstein manifold with a negative scalar curvature and has positive sectional curvature. It is also shown that a concircular flat normal metric contact pair manifold an Einstein manifold. Finally it is obtained that a quasi-conformal flat normal metric contact pair manifold is an Einstein manifold with a positive scalar curvature and, is a space of constant curvature.