Reduction and submanifolds of generalized complex manifolds

TL;DR

This paper explores the classical tensor field presentations of generalized complex, paracomplex, and subtangent structures, establishing their connections to Poisson geometry and extending reduction theorems to these structures.

Contribution

It introduces a unified tensor field framework for various generalized structures and proves reduction theorems and submanifold characterizations within this context.

Findings

Unified tensor presentation for generalized structures

Extension of Marsden-Ratiu and Marsden-Weinstein reduction theorems

Characterization of submanifolds inheriting induced structures

Abstract

We recall the presentation of the generalized, complex structures by classical tensor fields, while noticing that one has a similar presentation and the same integrability conditions for generalized, paracomplex and subtangent structures. This presentation shows that the generalized, complex, paracomplex and subtangent structures belong to the realm of Poisson geometry. Then, we prove geometric reduction theorems of Marsden-Ratiu and Marsden-Weinstein type for the mentioned generalized structures and give the characterization of the submanifolds that inherit an induced structure via the corresponding classical tensor fields.