Einstein almost cok\"ahler manifolds

TL;DR

This paper investigates Einstein almost cok"ahler manifolds, providing explicit examples and establishing conditions under which such manifolds are cok"ahler, along with bounds on their $*$-scalar curvature.

Contribution

It introduces the first explicit non-compact Einstein almost cok"ahler example and proves conditions for compact Einstein almost cok"ahler manifolds to be cok"ahler.

Findings

Explicit non-compact Einstein almost cok"ahler example

Compact Einstein almost cok"ahler manifolds with non-negative $*$-scalar curvature are cok"ahler

Bounds for $*$-scalar curvature in non-cok"ahler cases

Abstract

We study an odd-dimensional analogue of the Goldberg conjecture for compact Einstein almost K\"ahler manifolds. We give an explicit non-compact example of an Einstein almost cok\"ahler manifold that is not cok\"ahler. We prove that compact Einstein almost cok\"ahler manifolds with non-negative $*$-scalar curvature are cok\"ahler (indeed, transversely Calabi-Yau); more generally, we give a lower and upper bound for the $*$-scalar curvature in the case that the structure is not cok\"ahler. We prove similar bounds for almost K\"ahler Einstein manifolds that are not K\"ahler.