On the rigidity of the Sasakian structure and characterization of cosymplectic manifolds

TL;DR

This paper introduces generalized 'weak' metric structures on manifolds, extending classical contact and Sasakian geometries, and proves that Sasakian structures are rigid under these new definitions.

Contribution

It defines new 'weak' structures that broaden classical theories and proves the rigidity of Sasakian structures within this framework.

Findings

Weak Sasakian structures are homothetically equivalent to classical Sasakian structures.

Weak almost contact structures with parallel $\

A weak cosymplectic structure can be constructed on product manifolds.

Abstract

We introduce new metric structures on a smooth manifold (called "weak" structures) that generalize the almost contact, Sasakian, cosymplectic, etc. metric structures $(\varphi,\xi,\eta,g)$ and allow us to take a fresh look at the classical theory. We demonstrate this statement by generalizing several well-known results. We prove that any Sasakian structure is rigid, i.e., our weak Sasakian structure is homothetically equivalent to a Sasakian structure. We show that a weak almost contact structure with parallel tensor $\varphi$ is a weak cosymplectic structure and give an example of such a structure on the product of manifolds. We find conditions for a vector field to be a weak contact infinitesimal transformation.