On the geometry of compatible Poisson and Riemannian structures

TL;DR

This paper explores the geometric compatibility between Poisson and Riemannian structures on manifolds, introducing almost Kähler–Poisson manifolds via contravariant complex structures and analyzing their properties under symmetries.

Contribution

It introduces the concept of almost Kähler–Poisson manifolds using contravariant complex structures and studies their properties and symmetries.

Findings

Defined almost Kähler–Poisson manifolds.

Analyzed properties under structure-preserving maps.

Explored compatibility conditions between Poisson and Riemannian structures.

Abstract

We consider compatibility conditions between Poisson and Riemannian structures on smooth manifolds by means of a contravariant partially complex structure, or $f$-structure, introducing the notion of (almost) K\"ahler--Poisson manifolds. In addition, we study some of their properties under structure preserving maps and symmetries.