On the curvature of metric contact pairs

TL;DR

This paper studies the curvature properties of metric contact pairs, providing formulas for Ricci curvature, and classifies flat metric contact pair manifolds, showing flat non-Kähler Vaisman manifolds do not exist.

Contribution

It offers new curvature formulas for metric contact pairs and classifies flat cases, revealing limitations on the geometry of such manifolds.

Findings

Metrics associated to normal contact pairs cannot be flat.

Flat non-Kähler Vaisman manifolds do not exist.

Flat associated metrics occur only when characteristic leaves are at most three-dimensional.

Abstract

We consider manifolds endowed with metric contact pairs for which the two characteristic foliations are orthogonal. We give some properties of the curvature tensor and in particular a formula for the Ricci curvature in the direction of the sum of the two Reeb vector fields. This shows that metrics associated to normal contact pairs cannot be flat. Therefore flat non-K\"ahler Vaisman manifolds do not exist. Furthermore we give a local classification of metric contact pair manifolds whose curvature vanishes on the vertical subbundle. As a corollary we have that flat associated metrics can only exist if the leaves of the characteristic foliations are at most three-dimensional.