Generalized almost contact structures and generalized Sasakian structures

TL;DR

This paper introduces generalized almost contact and Sasakian structures on odd-dimensional manifolds, exploring their properties, examples, and equivalences, especially on compact 3-manifolds, expanding the geometric framework beyond classical structures.

Contribution

It defines generalized almost contact and Sasakian structures, provides examples on non-compact manifolds, and establishes equivalence results on compact 3-dimensional manifolds.

Findings

Generalized Sasakian structure on non-compact manifolds not arising from classical pairs.

On compact 3-manifolds, generalized Sasakian structures are equivalent to pairs of Sasakian structures with the same metric.

Extension of the definition of generalized almost contact structures.

Abstract

We introduce generalized almost contact structures which admit the $B$-field transformations on odd dimensional manifolds. We provide definition of generalized Sasakain structures from the view point of the generalized almost contact structures. We obtain a generalized Sasakian structure on a non-compact manifold which does not arise as a pair of ordinary Sasakian structures. However we show that a generalized Sasakian structure on compact 3-dimensional manifold is equivalent to a pair of Sasakian structures with the same metric. Finally we extend a definition of a generalized almost contact structure.